microstructural-mechanical property relationships in wc-co

Microstructural-Mechanical Property

Relationships in WC-Co composites

Chang-Soo Kim

Ph. D. Thesis

Materials Science and Engineering Department

Carnegie Mellon University

Pittsburgh, PA 15213, USA

Thesis Committee:

Dr. Gregory S. Rohrer (Advisor, Carnegie Mellon University)

Ted R. Massa (Co-advisor, Kennametal Incorporated)

Dr. Warren M. Garrison (Carnegie Mellon University)

Dr. Anthony D. Rollett (Carnegie Mellon University)

Dr. David M. Saylor (Food and Drug Administration)

Dr. Paul Wynblatt (Carnegie Mellon University)

Abstract

While empirical relationships between the mechanical properties of WC-Co

composites and the carbide grain size and carbide volume fraction are qualitatively well-

known, the influence of other microstructural features on the macroscopic properties has

been less clear. In this thesis, a comprehensive study of the effect of the interface

character distribution (grain shape and misorientation distribution of WC crystals), grain

size, size distribution, contiguity, and carbide volume fraction on the mechanical

properties (fracture strength) of WC-Co composites is described. To do this, methods

were developed to accurately measure microstructural features, characterize interfaces,

and predict mechanical properties.

Atomic force microscopy (AFM) and orientation imaging microscopy (OIM)

have been used to determine the WC/WC boundary and WC/Co interface positions in

WC-Co composites with Co volume fractions from 10 % to 30 % and mean carbide grain

sizes from 1 to 7 microns. Direct measurements of the carbide and cobalt area fractions

show that the log-normal distributions of carbide grain size and binder mean free path

distributions are Gaussian and that there is a very strong linear correlation between the

volume fraction of the binder and the contiguity of carbide phase. The number of vertices

per carbide grain is nearly constant and it is weakly influenced by the Co volume fraction.

Empirical expressions involving the grain size and carbide contiguity fit well to measured

hardness and fracture toughness data.

Stereological analysis of the carbide grain shapes shows that WC particles in

these materials have similar distributions of habit planes, with 101 0 prism facets and

the 0001 basal planes in contact with Co. The orientation imaging microscopy (OIM)

measurements demonstrate an absence of orientation texture and that the carbide grain

boundaries have similar misorientation distributions, with a high population grain

boundaries that have a 90° twist misorientation about the [101 0] axis and 30° twist and

asymmetric misorientations about the [0001] axis.

Using a two-dimensional finite element analysis (FEM) of the stress-strain field in

these materials, a brittle fracture model for the prediction of fracture strength has been

developed. The model assumes that a crack initiates either along carbide/carbide

boundaries or in carbide grains. When the fracture energy is ~49 J/m2, the calculated

fracture strength under a combined (thermal and mechanical) load shows good agreement

with the experimentally measured data. The calibrated model was then applied to the

hypothetical microstructures of WC-Co composites. The degree of carbide connectivity

(contiguity), orientation and/or misorientation texture appear to be the most important

parameter for improving the fracture strength of these materials.

Table of Contents

Chapter 1 – Introduction 1

1.1 Cemented carbides systems ------------------------------------------------------------- 1

1.1.1 History ------------------------------------------------------------------------------- 1

1.1.2 Classification ---------------------------------------------------------------------- 2

i) WC-Co grades ------------------------------------------------------------------ 2

ii) Alloyed grades ----------------------------------------------------------------- 2

1.1.3 Physical Properties --------------------------------------------------------------- 3

1.1.4 Applications ------------------------------------------------------------------------ 3

1.2 Motivations ---------------------------------------------------------------------------------- 4

1.3 Objectives ------------------------------------------------------------------------------------ 6

1.4 References ------------------------------------------------------------------------------------ 7

Chapter 2 – Background 8

2.1 Shapes of carbide particles -------------------------------------------------------------- 8

2.1.1 Anisotropy -------------------------------------------------------------------------- 8

2.1.2 Equilibrium shape of WC crystals in a Co binder ----------------------- 10

Wulff plot ---------------------------------------------------------------------------------- 11

2.2 Carbide/carbide boundaries ----------------------------------------------------------- 13

2.2.1 Basic crystallography ---------------------------------------------------------- 13

i) Orientation representation --------------------------------------------------- 13

Coordinate system transformations ------------------------------------------- 13

Fundamental zone (Standard triangle, unique triangle) ---------------------14

Pole figure (PF) ------------------------------------------------------------------ 16

Inverse pole figure (IPF) ------------------------------------------------------- 16

ii) Grain boundary representation --------------------------------------------- 16

Misorientations ------------------------------------------------------------------ 17

Misorientation distributions in Euler space ---------------------------------- 18

Misorientation distributions in axis-angle space ---------------------------- 19

Misorientations in Rodrigues space ------------------------------------------- 21

Tilt and twist boundaries ------------------------------------------------------- 21

Coincidence site lattice (CSL) boundaries ----------------------------------- 23

iii) Texture ----------------------------------------------------------------------- 24

Orientation texture -------------------------------------------------------------- 24

Misorientation texture ---------------------------------------------------------- 25

Grain boundary plane texture -------------------------------------------------- 26

2.2.2 Character of carbide/carbide grain boundaries --------------------------- 27

2.3 Measurements of microstructural parameters ----------------------------------- 28

2.3.1 Linear-intercept analysis ------------------------------------------------------- 28

2.3.2 Microstructural parameters ----------------------------------------------------29

i) Volume fraction --------------------------------------------------------------- 29

ii) Particle size and size distribution ------------------------------------------ 29

iii) Contiguity of carbide phase ------------------------------------------------ 30

iv) Angularity of carbide phase ------------------------------------------------ 31

2.4 Mechanical behavior of WC-Co composites ------------------------------------ 32

2.4.1 Elastic stress and stored energy ---------------------------------------------- 32

i) Stress invariant and strain invariant ---------------------------------------- 32

ii) Elastic energy density ------------------------------------------------------- 34

2.4.2 Elastic and plastic behaviors -------------------------------------------------- 34

i) Elastic constants -------------------------------------------------------------- 34

ii) Plastic deformation ---------------------------------------------------------- 36

2.4.3 Fracture approaches ------------------------------------------------------------ 37

i) Griffith condition ------------------------------------------------------------- 37

ii) Energy release rate and critical intensity factor -------------------------- 38

2.5 Microstructure-property relations of WC-Co composites ------------------- 39

2.5.1 Mechanical properties ---------------------------------------------------------- 39

i) Hardness ----------------------------------------------------------------------- 39

ii) Fracture toughness ----------------------------------------------------------- 40

iii) Transverse rupture strength (TRS) ---------------------------------------- 41

2.5.2 Past models for microstructure-property relations ----------------------- 42

i) Hardness ----------------------------------------------------------------------- 43

ii) Fracture toughness ----------------------------------------------------------- 44

2.6 Numerical models for structural analysis ----------------------------------------- 47

2.6.1 Finite element methods (FEM) for stress-strain calculations --------- 48

2.6.2 Previous analyses for the prediction of crack paths in WC-Co

composites using FEM ---------------------------------------------------------------- 51

2.6.3 Object-oriented finite element (OOF) model ----------------------------- 52

2.7 Summary ------------------------------------------------------------------------------------ 55

2.8 References ---------------------------------------------------------------------------------- 55

Chapter 3 – Microstructural Characterization 61

3.1 Samples -------------------------------------------------------------------------------------- 61

3.2 Microstructure and property characterization methods ---------------------- 65

3.2.1 Atomic force microscopy (AFM) ------------------------------------------- 65

3.2.2 Orientation imaging microscopy (OIM) ----------------------------------- 66

3.2.3 Microstructure analysis -------------------------------------------------------- 67

3.2.4 Measurements of mechanical properties ----------------------------------- 70

3.3 Results ---------------------------------------------------------------------------------------- 72

3.3.1 Grain size, binder mean free path, and size distributions -------------- 73

3.3.2 Contiguity of carbide phase --------------------------------------------------- 77

3.3.3 Carbide crystal shape ----------------------------------------------------------- 78

3.3.4 Orientation and misorientation distributions ------------------------------ 81

3.4 Correlation with mechanical properties ------------------------------------------- 82

3.4.1 Hardness --------------------------------------------------------------------------- 83

3.4.2 Fracture toughness -------------------------------------------------------------- 86

3.5 Summary ------------------------------------------------------------------------------------ 89

3.6 References ---------------------------------------------------------------------------------- 90

Chapter 4 – Analysis of Interfaces in WC-Co Composites 92

4.1 Samples -------------------------------------------------------------------------------------- 92

4.2 Stereological method for interface characterizations ------------------------- 93

4.2.1 Method overview ---------------------------------------------------------------- 93

i) Basic idea ---------------------------------------------------------------------- 93

ii) Trace classification ---------------------------------------------------------- 95

iii) Applicability and limitations of stereological analysis ----------------- 96

4.2.2 Numerical analysis -------------------------------------------------------------- 98

i) Two- and five-parameter calculations ------------------------------------- 98

ii) Number of line segments -------------------------------------------------- 101

iii) Background subtraction --------------------------------------------------- 103

iv) Data processing; OIM image clean-up ---------------------------------- 105

4.3 WC/Co interfaces ----------------------------------------------------------------------- 108

4.3.1 Analysis of grade E ------------------------------------------------------------ 108

i) Result from the manual tracings ------------------------------------------ 108

ii) Result from the automatic extractions ----------------------------------- 112

4.3.2 Analysis of other grades ------------------------------------------------------ 113

4.4 WC/WC boundaries -------------------------------------------------------------------- 116

4.4.1 Analysis of grade A (two-parameter calculation) ---------------------- 116

4.4.2 Analysis of other grades (two-parameter calculation) ----------------- 124

4.4.3 Five-parameter calculation -------------------------------------------------- 126

4.5 Summary ----------------------------------------------------------------------------------- 129

4.6 References --------------------------------------------------------------------------------- 129

Chapter 5 – Finite Element Simulations of Fracture Strength

in WC-Co Composites 132

5.1 Simulation methods -------------------------------------------------------------------- 132

Euler angles in OOF software ------------------------------------------------------------------ 138

5.2 Results from real microstructures using brittle fracture model ---------- 138

5.3 Results from hypothetical microstructures using brittle fracture model ----

------------------------------------------------------------------------------------------------------ 146

5.3.1 Contiguity ----------------------------------------------------------------------- 147

i) Hypothetical microstructures from simulations ------------------------- 147

ii) Hypothetical microstructures from real microstructures -------------- 153

5.3.2 Angularity ----------------------------------------------------------------------- 157


ii) Hypothetical microstructures from real microstructures of other

composites ---------------------------------------------------------------------- 160

5.3.3 Aspect ratio --------------------------------------------------------------------- 164


5.3.4 Size distribution ---------------------------------------------------------------- 167


5.3.5 Orientation and Misorientation --------------------------------------------- 170

i) Simple examples ------------------------------------------------------------ 170

ii) Hypothetical microstructures --------------------------------------------- 178

5.4 Summary ----------------------------------------------------------------------------------- 190

5.5 References --------------------------------------------------------------------------------- 190

Chapter 6 – Summary 192

6.1 References --------------------------------------------------------------------------------- 196

List of Figures

2.1 : Crystal structure of WC

2.2 : A typical optical micrograph of WC-Co composites

2.3 : Proposed equilibrium shapes of WC crystals

2.4 : Schematic illustration of Wulff plot and the corresponding Wulff construction

2.5 : Illustration of the laboratory coordinate system and crystal coordinate system

2.6 : Schematic representation of transformation from the laboratory coordinate to crystal

coordinate system

2.7 : A [0001] stereographic projection of hexagonal systems

2.8 : Example of misorientation distributions in Euler space

2.9 : Example of misorientation distributions in axis-angle space

2.10 : Example of misorientation distributions in Rodrigues space

2.11 : Schematic illustration of the pure tilt and twist boundaries

2.12 : Schematic illustration of CSL boundaries in superimposed simple cubic crystals

2.13 : Illustrations of orientation texture

2.14 : Illustrations of misorientation texture

2.15 : Illustration of grain boundary plane texture

2.16 : Schematic illustration of stress tensor. Only x (x1) and y (x2) components are

shown.

2.17 : Schematic plots of the relationship between the mechanical properties and the

microstructural parameters

2.18 : Schematic illustration of the 3-point transverse rupture strength (TRS) bend test

3.1 : AFM images of nine specimen grades

3.2 : Advantage of etching in Murakami’s reagent

3.3 : Typical images of AFM and OIM

3.4 : Example of direct measurements

3.5 : Plots of the carbide grain size distributions

3.6 : Carbide grain diameter distributions for each grade

3.7 : Comparison of the measured binder mean free path (direct measurements) and

calculated binder mean free path (stereology)

3.8 : Variation of the contiguity of carbide phase as a function of the carbide volume

fraction.

3.9 : Distributions of number of vertices and variation of average number of vertices

3.10 : Distributions of the number of vertices for each grade

3.11 : Comparison of the calculated hardness by empirical equation with experimentally

measured data.

3.12 : Comparison of the calculated hardness by empirical equation and the previous

models with experimentally measured data.

3.13 : Comparison of the calculated fracture toughness by empirical equation with

experimentally measured data.

3.14 : Comparison of the calculated fracture toughness by empirical equation and the

previous models with experimentally measured data.

4.1 : Schematic illustration of stereological analysis

4.2 : Schematic illustrations for the cross-sections of boundary structures

4.3 : Different types of boundaries as a function of resolution angles

4.4 : Illustration of discretized domain

4.5 : Variation of aspect ratio with the number of line segments included in the

calculations

4.6 : Example of automatic extraction of line segments

4.7 : WC/Co Habit plane probability distribution derived from 50 and 200 WC crystals of

grade E

4.8 : Comparison of WC crystal shape in AFM topography image and OIM inverse pole

figure (IPF) map

4.9 : The proposed average shape of WC crystal in Co

4.10 : Atomic models for the chemical termination of WC prism planes

4.11 : WC/Co interface plane distributions from automatic extractions

4.12 : Variation of aspect ratio with the carbide volume fraction.

4.13 : Pole figure of WC crystals

4.14 : Grain boundary population of WC crystals and random objects as a function of

misorientation angle

4.15 : WC/WC grain boundary misorientation distribution in axis-angle space

4.16 : Habit plane probability distribution for 90° boundaries and 30° boundaries of grade

A

4.17 : Schematic representations of the three most common types of grain boundaries in

composites

4.18 : Habit plane distributions of grad B from automatic extractions

4.19 : Grain boundary plane distributions using five-parameter calculations

5.1 : Example of OOF simulations

5.2 : Conversion of colors for OOF simulations

5.3 : Comparison of the calculated and measured moduli for each grade

5.4 : Stress invariant 1 and elastic energy density distributions (from grade A) under a

combined load

5.5 : Comparison of the calculated and measured strengths for each grade

5.6 : Examples of hypothetical microstructures (for contiguity) with their stress invariant

1 and elastic energy density distributions under a mechanical load

5.7 : Average stresses in the composites, carbide and binder phases with different

contiguities

5.8 : Plot of the calculated fracture strength with different contiguities

5.9 : Examples of hypothetical microstructures (for contiguity) with their EED

distributions

5.10 : Examples of hypothetical microstructures (for contiguity) with their elastic energy

density distributions

5.11 : Schematic illustration of stress distributions of high and low contiguity

microstructures

5.12 : Examples of hypothetical microstructures (for angularity) with their SI1 and EED

distributions

5.13 : Plot of the calculated fracture strength with different angularities

5.14 : Examples of hypothetical microstructures (for angularity) with their EED

distributions

5.15 : Schematic illustration of the stress concentration at sharp edge of WC phase in a y-

elongation simulation

5.16 : Examples of hypothetical microstructures (for aspect ratio) with their SI1 and EED

distributions

5.17 : Plot of the calculated fracture strength with different aspect ratios

5.18 : Schematic illustration of the stress concentrations with different aspect ratios in a

y-elongation simulation

5.19 : Examples of hypothetical microstructures (for size ratio) with their EED

distributions

5.20 : Plot of the calculated fracture strength with different size ratios

5.21 : Schematic illustration of the stress concentrations with different size ratios in a y-

elongation simulation

5.22 : Schematic illustration of (001)C//[001]S orientation texture samples (pseudo-cubic

materials) with different misorientations

5.23 : SI1 distribution of [001] orientation texture samples (pseudo-cubic materials) with

different misorientations

5.24 : EED distribution of [001] orientation texture samples (pseudo-cubic materials)

with different misorientations

5.25 : Effects of orientation and misorientation distributions on stress and elastic energy

density distributions

5.26 : Stored energy density variation with orientation and misorientation textures

5.27 : Estimated fracture strength in hypothetical microstructures with and without

textures

5.28 : deviation angle (∆) between the [001] sample axis and a <0001> crystal axis

5.29 : Probability of finding a crystal as a function of texture intensity factor (β) and

deviation angle (∆) from a sample axis

5.30 : Calculated modulus with different orientation textures (from grade A)

5.31 : Schematic illustration of the orientation in [001] texture and [100] texture

5.32 : SI1 distributions with different orientation textures (from grade A)

5.33 : EED distributions with different orientation textures (from grade A)

5.34 : The calculated fracture strength and fracture strength increase with different

orientation textures

5.35 : The calculated fracture strength increase in the hypothetical microstructures of

grades A, B, C, and E

5.36: The calculated maximum fracture strength increase as a function of contiguities

6.1 : Schematic illustration of the effects of microstructural features on the fracture strength of composites.

List of Tables

3.1 : Carbide volume fractions for each specimen grade

3.2 : Mechanical properties for each grade

3.3 : Summary of the results of microstructures

5.1 : Elastic constants of WC and Co

5.2 : Summary of the microstructural and mechanical properties for grades A, B, C, and E

Chapter 1

Introduction

Cemented carbides (or sintered carbides) are common hard materials which have

outstanding mechanical properties that make them commercially useful in machining,

mining, metal cutting, metal forming, construction, wear parts, and other applications [1-

3]. Since the early 20th century, the cemented carbides have been widely used in many

manufacturing processes that benefit from their combination of high hardness, fracture

toughness, strength, and wear resistance. This chapter provides a brief overview of

cemented carbide systems, describes the motivation for the current work, and defines the

objectives of this thesis.

1.1 Cemented carbide systems

1.1.1 History

Tungsten monocarbide (WC, usually referred to as tungsten carbide) was

discovered by Henri Moissan in 1893 during his search for a method to make synthetic

diamonds [4]. He found that the hardness of WC is comparable to that of diamond. This

material, however, proved to be so brittle that its commercial use was seriously limited.

Subsequently, research was focused on improving its toughness, and significant

contributions to the development of cemented carbides were made in the 1920’s by Karl

Schröter [4]. Employing cobalt (Co) as a binding material, Schröter developed a

1

compacting and sintering process for cemented tungsten carbides (WC-Co) that is still

widely used to manufacture WC-Co composites. Most of the further developments were

modifications of the Schröter’s process, involving replacement of part or all of the WC

with other carbides, such as titanium carbide (TiC), tantalum carbide (TaC), and/or

niobium carbide (NbC).

1.1.2 Classification

i) WC-Co grades

Straight grades, sometimes referred to as unalloyed grades, are nominally pure

WC-Co composites. They contain 3-13 w/o (weight percent) Co for cutting tool grades

and up to 30 w/o Co for wear resistant parts. The average carbide particle size varies

from sub-micron to eight microns. Straight grades are used for machining cast iron,

nonferrous alloys, and non-metallic materials, but generally not for the machining of steel.

This thesis is focused on the study of straight grades.

ii) Alloyed grades

Alloyed grades are also referred to as steel cutting grades, or crater resistance

grades, which have been developed to prohibit cratering during the machining of steel.

The basic compositions of alloyed grades are 3-12 w/o Co, 2-8 w/o TiC, 2-8 w/o TaC,

and 1-5 w/o NbC. The average carbide particle size of these grades is usually between 0.8

and 4 µm. In this thesis, only a single alloyed grade was studied, as an example of a

microstructure that is significantly different from the straight grades.

2

1.1.3 Physical Properties

The physical properties of these composites depend on microstructural features,

such as grain sizes, size distributions, grain shapes, orientations, misorientations, and the

volume fraction of the carbide phase. The hardness, toughness, and fracture strength of

WC-Co composites range from 850 to 2000 kg/mm2 (Vickers hardness, HV), from 11 to

25 MPa (critical stress intensity factor, plane strain fracture toughness, KIC), and from 1.5

to 4 GPa (transverse rupture strength, TRS), respectively. Also, it is known that the wear

resistance of these materials is five to ten times higher than that of a typical tool steel.

The details of the physical properties of WC-Co composites will be described in Chapter

2. It should be noted at the outset that while the relationships between the mechanical

properties and the mean grain size, carbide volume fraction are known, the influence of

grain shape, size distribution, and interface character distribution are not yet clear.

Furthermore, it is not clear how changing these microstructural characteristics beyond

normally observed ranges alters the properties of the composites.

1.1.4 Applications

The cemented carbides are primarily used as cutting tools. The most wear

resistant materials are the straight WC-Co grades which are used for machining and

cutting of materials that require abrasion resistance; e.g. cast iron and nonferrous alloys.

On the other hand, the alloyed grades are used for strongly yielding materials that require

high deformation resistance, toughness, and thermal shock resistance. Also, many

applications exist for cemented carbides in non-cutting areas such as mining, construction,

and wear resistance components (for example, various types of dies, nozzles, and rolls).

3

In these cases, the composites are designed to meet the demands of the specific

applications which may require impact, abrasion, and/or corrosion resistance.

1.2 Motivations

Like many other engineering materials, the mechanical properties of cemented

carbides are strongly influenced by their microstructures. While the qualitative effects of

microstructural variables on the mechanical performance are relatively well-understood,

the quantitative functional relationships between micro- and macro-features and their

fundamental underpinnings are not well-established. This work focuses on a

comprehensive analysis of relationships between microstructural and mechanical

properties. As microstructural parameters, we include the interface character distribution

in WC-Co composites, as well as the conventional factors, such as grain size, size

distribution, carbide volume fraction, and the carbide connectivity (contiguity). As a

mechanical property parameter, we mainly focus on the fracture strength (transverse

rupture strength, TRS) of composites. Only the modeling of fracture strength is included

in this thesis because, currently, it is not feasible to numerically approximate other

mechanical properties such as hardness and fracture toughness. For the comprehensive

understanding of the relationships, it is necessary to develop methods for the accurate

measurements of microstructures, characterization of interfaces, and prediction of

fracture strengths.

Conventionally, the linear-intercept scheme (an indirect method) has been used to

characterize microstructural features in WC-Co composites. This methodology inevitably

4

involves inaccuracies and limitations in measuring microstructural parameters (see

Section 2.3.1 for details). Especially, the linear-intercept method is not able to evaluate

the shape of the carbide phase and its angularity. To understand the fundamental

interrelations of micro- and macro-features, it is important to establish an accurate and

direct microstructural measurement method. This will be a foundation for developing a

model that allows one to predict mechanical properties.

A second issue is that most of the previous microstructural measurements have

been orientation averaged. In other words, the anisotropic distributions of misorientations

and grain boundary planes of these composites have not been explored. To develop a

comprehensive description of the microstructure and, to understand the three dimensional

shape, a quantitative analysis of interface crystallography (carbide/carbide boundaries,

binder/carbide interfaces) is required. During the processing of materials, the

thermodynamic driving force for the formation, dissolution, and evolution of

microstructures is mainly provided by the excess free energies of interfaces. These

energies are, therefore, closely related to the materials performance. However, little is

known about the interface crystallography of WC-Co composites. In the case of WC/Co

interface character, the equilibrium shape of WC in Co is still the matter of controversy

(see Section 2.1.2 for details). Much less is known about WC/WC boundary character

distribution, but some transmission electron microscopy (TEM) observations of special

boundaries have been reported [5].

Finally, a model that can be used to calculate fracture strengths of hypothetical

microstructures would be valuable for the development of new carbide materials. Since

the microstructures of hypothetical materials could be controlled to produce nearly

5

independent variations in the microstructural parameters, the effects on transverse rupture

strength (TRS, or equivalently, fracture strength) would be clearly revealed. To test the

effects of individual microstructural features, we first have to establish a reliable model to

approximate the fracture strength using real microstructures. Although several models

have been proposed to simulate fracture mechanisms of WC-Co composites, the

application of these previous models has been limited to the calculations of stress-strain

distributions in relatively small areas, and the crystallographic anisotropies were not

incorporated (see Section 2.6.2 for details). Thus, the development of a reliable model to

predict the fracture strength prediction remains a challenge.

1.3 Objectives

Considering the challenges outlined above, the focus of this work is on the

quantitative and comprehensive analysis of microstructural features, interface

characteristics, and mechanical properties (fracture strength) of straight WC-Co grades.

This thesis has the followings goals:

1) Establish a reliable method for the accurate measurements of microstructural features

of WC-Co composites. From this, specify robust functional relationships between

microscopic parameters and macroscopic physical properties.

2) Develop a method to analyze the anisotropic distribution of interfaces in WC-Co

composites. Then, quantify the distribution of interface types (with respect to the lattice

misorientation and boundary plane orientation) in WC-Co composites.

6

3) Develop a numerical simulation model to predict the fracture strength as a function of

quantifiable characteristics of microstructures, and propose WC-Co composite

microstructures with improved fracture strengths.

1.4 References

[1] H.E. Exner, “Physical and Chemical Nature of Cemented Carbides”, Int. Metal. Rev.,

4, 149 (1979)

[2] J. Gurland, “New Scientific Approaches to Development of Tool Materials”, Int.

Mater. Rev., 33, No.3, 151 (1988)

[3] B. Roebuck and E.A. Almond, “Deformation and Fracture Processes and the Physical

Metallurgy of WC-Co Hardmetals”, Int. Mater. Rev., 33, No.2, 90 (1988)

[4] G. Schneider, Jr., Principles of Tungsten Carbide Engineering, 2nd ed., Society of

Carbide and Tool Engineers, 1989

[5] S. Hagege, G. Nouet, and P. Delavignette, “Grain Boundary Analysis in TEM (IV)”,

Phys. Stat. Sol. A, 61, 97 (1980)

7

Chapter 2

Background

This chapter contains background information relevant to this thesis. First, a brief

overview of interfaces in WC-Co composites is given. Next, the basic crystallography

and microstructures of the composites are described. This is followed by a short review

of the mechanical properties, fracture mechanisms and empirical microstructure-property

relationships for the composites. Finally, finite element models (FEM) for stress-strain

field calculations, with their capabilities and applications, are presented.

2.1 Shapes of carbide particles

2.1.1 Anisotropy

Cemented carbides are sintered in the presence of a liquid Co phase. During

processing, the characteristic carbide microstructure is formed by coarsening. In the

coarsening, WC dissolves from relatively smaller crystals, diffuses through the liquid Co,

and precipitates on relatively larger crystals. Both the interfacial energies and

attachment/detachment at the solid (WC)/liquid (Co) interface affect the shapes of the

WC crystals that grow and shrink during the coarsening process.

Interfacial energy can be defined as the work that must be done to create a unit

area of an interface at constant T, P, and µ (temperature, pressure, and chemical potential,

respectively). In solid materials, the interfacial energy is affected by the crystal structure

8

(number of broken bonds, interplanar spacing, and the charge balance of cations and

anions in ionic compounds). While the symmetry of the interfacial energy is normally

same as that of the crystal structure, surface polarity can generate additional anisotropy

[1]. WC has a hexagonal crystallographic structure with two atoms (W and C) per unit

cell as illustrated in Fig. 2.1. The lattice constants are a=2.906 Å and c=2.837 Å with

c/a=0.976 [2, 3].

120°

c-axis (2.84Å)

a-axis

(2.91Å)

Fig. 2.1 : Crystal structure of WC. Blue and green denote W and C atoms, respectively. W atoms are at (0, 0, 0) and C atoms are at (1/2, 2/3, 1/2) positions in the WC unit cell.

A typical optical micrograph of a straight grade WC-Co composite is shown in

Fig. 2.2 (10 Co v/o, grade A in Chapter 3-5). It is clear that the WC surfaces are faceted

when WC is embedded in Co. While alloyed cemented carbides frequently display more

spherical (or rather rough) WC grains, WC particles in straight WC-Co grades invariably

exhibit angular shapes. Based on this angular character of WC particles, it is thought that

the WC/Co interfaces possess strong anisotropies and WC crystals have well-defined

habit planes.

9

Fig. 2.2 : A typical optical A in Chapter 3-5). Dark ye contrast denotes Co phase. In

e are two different types of interfaces: WC/Co interface

ibrium shape of WC crystals in a Co binder

The carbide/carbide and binder/carbide interfacial energies have not yet been

n reported that the liquid Co

comple

atic

micrograph of WC-Co composites (from specimen grade llow contrasts represent WC grains and bright yellow

20 µm

composites, ther and WC/WC grainboundaries.

2.1.2 Equil

quantitatively measured. By wetting experiments, it has bee

tely wets the WC crystals [4-6], indicating that the average binder/carbide

interfacial energy is much lower than the average carbide/carbide interfacial energy.

Previously, from observations of deeply etched samples, the shapes of WC

particles in a Co binder have been identified both a trigonal prism with three prism

10

and two basal facets (see Fig. 2.3(a)), or a truncated trigonal prism where some of the

corners or edges of the trigonal prism are replaced by new facets (see Fig. 2.3(b)) [4, 7-

10]. From these results, the (0001) basal and )0110( prismatic facets of the carbide phase

are considered to have relatively low interface energy in contact with Co. In all of the

cases of the past studies, however, the shape C in a Co binder were derived from

qualitative observations. Thus, the quantitative aspect ratios (base-to-height ratio of prism

facet) of the crystals have not been measured. Furthermore, since deep etching can not

reveal the full three dimensional shape, it is impossible to know which faces were

actually created by impingement at a part of the crystal already removed by polishing or

below the level of the etch.

s of W

with truncated

edge shape.

If a particle (or a grain) adopts the equilibrium shape, a Wulff plot (or γ plot) can

estimate WC/Co interfacial energy anisotropies. In a Wulff plot (see Fig.

Fig. 2.3 : Proposed equilibrium shapes of WC crystals. (a) Prism shape, (b) prism

(a) (b)

Wulff plot

be used to

11

2.4(a)),

the direction of the vector to a specific point on the polar plot represents the

orientation of the interface, and the magnitude of the vector indicates the relative value of

the interfacial energy at that particular orientation. The equilibrium shape is given by the

inner envelope of planes perpendicular to each orientation vector (the shaded area in Fig.

2.4(b)). Highly angular WC particle shapes indicate that there should be deep cusps

(singularities) in the WC/Co interfacial energy contour, and that there are missing

orientations in the WC equilibrium form. It is not clear that the WC crystals in WC-Co

composites are equilibrium shapes. The observed WC shapes in these composites are

influenced by the mechanisms of dissolution/growth that occur during coarsening and by

the impingements of carbide crystals.

(a) (b) Fig. 2.4 : (a) Schematic illustration of Wulff plot. Surface energy vector, γ(θ), represents the

surface energy at a specific orientation θ. (b) The corresponding Wulff construction. The tors is the equilibrium

relativeinner envelope (shaded area) formed by the normals to surface energy vecshape.

12

2.2 Carbide/carbide boundaries

ations

Physically, there are two reference frames to describe the orientations of grains in

sample) reference frame and crystal reference frame

(see Fig

2

rma

2.2.1 Basic crystallography

i) Orientation representation

Coordinate system transform

polycrystals. They are laboratory (or

. 2.5, laboratory frame in red, crystal frame in blue). In this document, interfaces

will be represented in the crystal reference frame. To transform the laboratory coordinate

frame to the crystal coordinate frame, we use three Euler angles (φ1, Φ, φ2) which define

a sequence of rotations from one reference frame to the other reference frame. If ix

denote a basis unit vector in laboratory coordinates, we need three consecutive rotations

( ix 'îx ''îx '''îx , where '''îx denote a basis unit vector in crystal coordinates, and

'îx and ''îx stand for a basis unit vector in some intermediate coordinates between the

laboratory and crystal frames) using the three Euler angles (see Fig. 2.6); ˆ yields 'ˆ by

φ about x axis, 'x yields ''x by a rotation of Φ about 'x axis, and ''x

yields '''ˆ by a rotation of φ about ''ˆ axis. Here, the Euler angle notation is that of

Bunge’s [11]. Using these three successive rotations, it is possible to construct one

transfo tion matrix (g), which converts the laboratory coordinate system to crystal

coordinate system. It is given by [11]:

ix ix

a rotation of 1 3 i i 1 i

ix 3x

13

⎤

⎢⎢

⎡−−

+−=

Φφφφφφ

ΦφΦφφφφΦφφφφφΦφ

ssccssc

sscsccscssccg

),,( 2121

221212121

21

⎥⎥⎥

⎦

⎢

⎣ −+−

ΦΦφΦφΦφφφφΦ

cscsccccss

11

22121 (2.1)

c and s represent cosine and sine, respectively.

Fig. 2.5 : Ill tal coordinate system

(blue). represent unit vectors in the laboratory and crystal reference frames, respectively.

enta

If we ignore the multiplicity of a special position (a position which contains

symmetry elements such as rotation axis, mirror plane, and/or inversion center), there are

24 symmetrically equivalent orientations in a hexagonal crystal system (e.g.

point group) [12]. In case of a cubic crystal system (e.g.

where

ustration of the laboratory coordinate system (red) and crys'''ˆ,ˆ ii xx

Fundam l zone (Standard triangle, unique triangle)

mmm/6

mm3 point group), the number

of equivalent orientations of a general position (a position that does not lie on a symmetry

element) increases to 48 [12]. Because of symmetry, any orientation can be described in a

fundamental zone (the smallest unit of a crystal system that contains all distinguishable

orientations). The fundamental zone of the hexagonal system is given in the [0001]

1x2x

3x

'''ˆ1x

'''x'''3x2

14

stereographic projection shown in Fig. 2.7. The choice of a fundamental zone is arbitrary,

and, in this work, the shaded region will be used as the fundamental zone of WC crystal.

(c) Fig. 2.6 : Schematic representation of transformation from the laboratory (red) coordinate to

blue) coordinate system. (a) Rotation from the laboratory coordinate system, , to a new

em (purple),

(a) (b)

crystal ( ix

coordinate system, 'îx (pink), (b) rotation from 'îx to another new coordinate syst , ''îx

(c) rotation from ''îx to the crystal coordinate system, '''îx .

Fig. 2.7 : A [0001] stereographic projection of hexagonal systems. The shaded area represents an example of the fundamental zone.

33

'ˆˆ xx =

φ

2x

'ˆ1x

'ˆ2x

'x3

''ˆ2x''ˆ3x

φ

Φ 'ˆ2x

'''ˆ''ˆ 33 xx =

'''ˆ1x

'''ˆ2x

''ˆ1x

''ˆ2x

1x ''ˆ'ˆ 11 xx =

15

Pole figure (PF)

Crystal orientations are usually plotted as two-dimensional stereographic

projections in pole figures (PF). A pole figure (PF) shows the position of a crystal pole

boratory reference frame. For example, in a (0001) pole figure of a

polycry

e position of a sample direction relative to

e crystal reference frame. Since the IPF has the symmetry of the crystals, it can be

one. Pole figures, on the other hand, have only the symmetry of

the sam

the type or character of a grain boundary, five independent

acroscopically observable parameters are required. The first three describes the

relative to the la

stal with a hexagonal symmetry, the axes on the stereographic projection indicate

directions in the laboratory coordinate system, and the positions of poles represent

vectors normal to the (0001) plane of a given crystal with respect to the laboratory

reference frame. Such pole figures are usually used for the analysis of the crystal

orientation distributions (orientation texture).

Inverse pole figure (IPF)

An inverse pole figure (IPF) shows th

th

plotted in a fundamental z

ple (usually taken to be an orthorhombic system) and are typically plotted on the

complete stereographic projection of a hemisphere. In a (0001) IPF, the indices in a

fundamental zone describe a direction in the crystal coordinate system, and the positions

of poles denote the crystal direction (in the crystal reference frame) that is aligned with

the sample normal.

ii) Grain boundary representation

To describe

m

16

misorie

The lattice misorientation (∆gab) across the boundary is calculated from the

individual adjoining crystals (ga and gb). To bring crystal b into

coincid

ggg =∆ (2.2)

T gb. From

and the vector components of misorientation (ni) can be calculated by following

ntation of a bicrystal and the last two identifies the grain boundary plane. For

numerical calculations used in this research, the lattice misorientation is specified by the

three Euler angles (φ1, Φ, φ2), and the two parameters that determine the boundary planes

are specified by the two spherical angles (φ, θ). Other possible choices for the

misorientation representations are the axis-angle descriptions (either in an axis-angle

space or in a Rodrigues space), which will also be used to display and discuss the results.

The grain boundary distribution is usually represented by the notation λ(∆g, n), which is

the relative occurrence of a grain boundary with a misorientation, ∆g, and a grain

boundary plane normal, n, in a unit of multiples of a random distribution (MRD).

Misorientations

orientations of

ence with crystal a, two successive transformations are specified; first, crystal b is

transformed into laboratory coordinates (gbT), and then transformed to the frame of

crystal a (ga). Therefore, the overall transformation matrix becomes [11]:

baabT

where gb denotes the transpose of matrix Eq. (2.2), the misorientation angle (ω)

equations [11]:

⎟⎠⎞

⎜⎝⎛ −

= −

21

cos 1 iig∆ϖ (2.3)

17

ω

∆εsin2

jkijk g−=in (2.4)

where ε is the permutation tensor.

space

Misorientations can be represented in terms of three Euler angles (

ijk

Misorientation distributions in Euler

21 ,, φ∆∆Φφ∆ ).

the consecutive rotation angles to transform

from a n example

⎦⎢⎣ − ∆Φ∆Φφ∆∆Φφ∆ cscss 11

221212121

here c and s represent cosine and sine, respectively.

Here, the three Euler angles are defined by

crystal reference frame to another crystal reference frame. A of

misorientation distributions in WC/WC grain boundaries (specimen grade A, Chapter 3-5)

using the Euler space is shown in Fig. 2.8. Variations of WC/WC boundary

misorientation distributions along ∆φ1 and ∆Φ are illustrated in one square section with a

specific ∆φ2. Distribution scales are represented as multiples of a random distribution

(MRD). The distribution in Fig. 2.8 is for the visualization purpose, not for the analysis

of the data (Fig. 2.9 and 2.10 also for the descriptions of axis-angle and Rodrigues spaces

only). The overall transformation matrix, ∆g, is same as Eq. (2.1), except the ∆ notations:

⎥⎥⎤

⎢⎢⎡

+−−−+−

= ∆Φφ∆∆Φφ∆φ∆φ∆φ∆∆Φφ∆φ∆φ∆φ∆∆Φφ∆∆Φφ∆φ∆φ∆φ∆∆Φφ∆φ∆φ∆φ∆

∆ sccccssccsscsscsccscsscc

g

221212121 (2.5)⎥

w

18

Fig. 2.8 : Example of misorientation distributions in Euler space (from grade A, Chapter 3-5).

Scales are multiples of a random distribution (MRD). ∆φ2 is specified at each section.

Misorientation distributions in axis-angle space

Another way to describe misorientation distributions is to use misorientation axis-

angle pairs. Any misorientation can be specified by a set of the smallest misorientation

angle (disorientation angle, Eq. (2.3)) and misorientation axis (Eq. (2.4)). Because every

misorientation can be represented by the crystallographically equivalent axis-angle pair

with the minimum misorientation angle, the maximum misorientation angle is bounded;

19

the upper limit depends on the crystal system. For example, the maximum misorientation

angles are ~63° in a cubic system and ~98° in a hexagonal system. In Fig. 2.9, an

example of misorientation distributions for WC/WC boundaries (same data set used in

Fig. 2.8, data from grade A in Chapter 3-5) is illustrated in axis-angle space. Each

standard triangle contains the misorientation axis distributions at a specific misorientation

angle.

Fig. 2.9 : Example of misorientation distributions in axis-angle space (from grade A, Chapter 3-5). Scales are multiples of a random distribution (MRD). Misorientation angle is specified at each section.

20

Misorientations in Rodrigues space

The representation of misorientation in Rodrigues space was first considered by

Heinz and Neumann [12], followed by Morawiec and Field [13]. A Rodrigues vector is

defined by:

ii nr ⎟⎠⎞

⎜⎝⎛=

2tan ϖ (2.6)

where ri is the Rodrigues vector, ω is the misorientation angle, and ni is the

misorientation axis. Euler angle (in Euler space) and Rodrigues vector (in Rodrigues

space) parameterizations are useful for the representations of orientation distributions as

well as misorientation distributions. An example of misorientation distributions of

WC/WC boundaries (same data set used in Fig. 2.8 and Fig. 2.9, data from grade A in

Chapter 3-5) is depicted in Fig. 2.10. Here, x, y, and z axes (in Cartesian coordinate)

correspond to the r1, r2, and r3 vectors, respectively.

Tilt and twist boundaries

A boundary is commonly described in terms of tilt and twist components. A tilt

boundary is defined as a boundary in which the misorientation axis lies in the boundary

plane and a twist boundary is one in which the misorientation axis is perpendicular to the

boundary plane. In real microstructures, most gain boundaries have mixed characteristics

of pure tilt and twist components. Examples of pure tilt and twist boundaries are

illustrated in Fig. 2.11 (a) and (b).

21

r2

r1

Fig. 2.10 : Example of misorientation distributions in Rodrigues space (data from grade A, Chapter 3-5). Scales are multiples of a random distribution (MRD). Magnitude of r3 vector is specified at each section.

(a) (b) Fig. 2.11 : Schematic illustration of the pure tilt and twist boundaries. (a) Pure tilt, (b) pure twist. The pink arrow represents the misorientation axis.

0.7 1.0 1.4 1.9 2.6 3.5 4.9 6.7

22

Coincidence site lattice (CSL) boundaries

When a finite fraction of lattice sites coincide between the two adjacent crystal

lattices, then the boundary is called as a coincidence site lattice (CSL) boundary. It is

widely believed that this coincidence leads to a reduction in the interfacial energy,

especially in the case of high angle boundaries. However, this hypothesis is not always

consistent with observations. The CSL boundaries are described by a quantity Σ, the

reciprocal ratio between the areas enclosed by a unit cell of the coincidence sites; as the Σ

value decreases, the fraction of coincident sites increases. This occurs only for special

boundary planes, and only in these cases, we do expect any effect on the grain boundary

energy. Fig. 2.12 illustrates examples of the pure twist Σ5 and Σ13 CSL boundaries in a

simple cubic (SC) crystal.

(a) (b) Fig. 2.12 : Schematic illustration of CSL boundaries in superimposed simple cubic crystals. (a)

Twist Σ5 boundary (misorientation angle=36.86°), (b) Twist Σ13 boundary (misorientation

angle=22.62°). The orange square in each figure leads a low Σ CSL boundary.

23

iii) Texture

The texture, Λ(g, ∆g, n), is defined as the relative frequency of occurrence of a

orientation, g, misorientation, ∆g, and/or grain boundary plane normal, n, in units of

multiples of a random distribution (MRD). If there is a preferred crystal orientation with

respect to sample reference frame, it is usually referred to as a texture. Preferred

misorientations with respect to the bicrystal reference frame are called misorientation

textures. And, preferred grain boundary planes with respect to the crystal reference frame

are called grain boundary plane textures. Schematic illustrations of these three textures

are presented in Fig. 2.13~15. We shall see that there are strong misorientation and grain

boundary plane texture in WC/WC boundaries of WC-Co composites (see Chapter 4).

Orientation texture

Orientation texture is described by the preferred orientations of crystals with

respect to sample reference frame. It is usually specified by the orientation distribution

function (ODF), Λ(g), in units of MRD. Fig. 2.13 (a) and (b) represent the

microstructures without and with orientation textures when the colors and arrows denote

the orientations and a certain plane normal direction type (for example, <110>) of

individual grains, respectively. It is known that this orientation texture can reduce the

stress and stored energy distributions in polycrystalline materials under a thermal load.

The effect of orientation texture on the strength of WC-Co composites will be discussed

in Section 5.4.3 ii).

24

(a)

(b) Fig. 2.13 : Illustrations of orientation texture (Λ(g)). Grain colors correlate with orientations. (a) Polycrystals without orientation texture, (b) polycrystals with orientation texture.

Misorientation texture

If grain boundaries tend to have specific misorientations (specific misorientation

axis-angle pairs), then the microstructures are said to have misorientation textures. It is

commonly described by the misorientation distribution function (MDF), Λ(∆g), in units

of MRD. Examples of the misorientation texture in WC-Co composites are given in Fig.

2.8~2.10. In those figures, high MRD values are found at specific misorientations. There

can be misorientation textures regardless of the existence of orientation textures. If the

red bold lines in Fig. 2.14 (b) and (c) indicate a specific misorientation (for example, 60°

about <111>), then the polycrystals show microstructures with misorientation textures, (b)

without misorientation texture, and (c) with orientation texture.

25

(a)

(b)

(c) Fig. 2.14 : Illustrations of misorientation texture (Λ(∆g)). Red grain boundaries denote a specific same misorientation. (a) Poly crystals with no orientation, no misorientation texture, (b) polycrystals with no orientation, but with misorientation texture, (c) polycrystals with orientation and misorientation texture.

Grain boundary plane texture

While misorientation textures in a polycrystalline material identify the existence

of preferred misorientations, grain boundary plane textures can be regarded as the

presence of preferred boundary planes between adjacent crystals in a polycrystal. The

grain boundary texture, Λ(n), is only the function of a grain boundary plane normal, n.

Also, grain boundary texture can develop in the absence of orientation and misorientation

textures. An example of a grain boundary plane textured polycrystal (without orientation

26

texture) is illustrated schematically in Fig. 2.15, where the red grain boundary planes

stand for a specific type of orientation (for example, 100).

Fig. 2.15 : Illustration of grain boundary plane texture (Λ(n)). Red grain boundaries have a specific orientation, hkl, in the crystal reference frame. Grain boundary texture can develop in the absence of orientation and misorientation texture.

2.2.2 Character of carbide/carbide grain boundaries

In WC-Co composites, there are binder/carbide interfaces and carbide/carbide

grain boundaries. Therefore, one can assume that the microstructure consists of a

contiguous carbide phase and an isolated binder phase. This is the so called ‘skeletonized

carbide’ model [15]. However, it has been reported that carbide/carbide boundaries

invariably contain cobalt as a submonolayer segregant implying that carbide particles are

dispersed in a continuous binder phase. In the current literature, this is referred to as the

‘dispersed carbide’ model [16, 17]. Whether the carbide/carbide boundary has thin binder

layer or not, little is known about the crystallographic character of carbide/carbide

boundaries. From the transmission electron microscopy (TEM) analysis, Hagege et al.

[18] observed a 90° twist boundary about [101 0] , which is related to a low Σ CSL

boundary (near Σ2). Based on the inhomogeneous distribution of observed dihedral

angles between contiguous WC grains, Deshmukh and Gurland [19] proposed that certain

grain boundaries are not penetrated by Co and, therefore, occur with a higher frequency.

27

Although they only examined only 10 carbide/carbide boundaries, they showed,

qualitatively, a tendency toward high coincidence contacts between carbide particles (7 of

10 carbide/carbide boundaries were low Σ CSL boundaries). One of the goals of this

thesis is to quantitatively specify the relative population of the carbide/carbide grain

boundaries.

2.3 Measurements of microstructural parameters

2.3.1 Linear-intercept analysis

In past studies, three parameters have been used to characterize the microstructure

and correlated to the mechanical properties of WC-Co composites; the volume fraction of

each phase, the average particle size of each of the phases, and the contiguity of carbide

phase. In addition to these three common microstructural parameters, the angularity of

the carbide phase is also considered to be a crucial factor. Conventionally, these

microstructural parameters have been characterized by a combination of point and linear

analysis (the linear-intercept analysis method) on scanning electron microscopy (SEM)

images. Although the linear-intercept method has the merits of providing information on

the volume fraction, size distribution, and contiguity, it also has several limitations. Most

importantly, grain shape and area cannot be estimated by this method. And, for

statistically reliable results, on the order of 103 observations need to be recorded and

analyzed. Finally, it is not possible to specify the angularity of the WC phase by the

linear-intercept method.

28

2.3.2 Microstructural parameters

i) Volume fraction

It is a common practice to specify the composition of WC-Co hard materials in

terms of weight percent (w/o), but the use of volume percent (v/o) is more informative,

since some of the W and C atoms are dissolved in the Co binder [20]. The densities of

Co-W-C solid solutions vary from 8.8 to 9.5 g/cm3 with W and C contents. The volume

fraction of each phase is the most important factor for determining the mechanical

properties of WC-Co composites. As the volume fraction of the carbide phase increases,

hardness increases and fracture toughness decreases. The volume fractions of the carbide

and binder phase can be obtained from random point counting with at least 103 counting

points.

cb

cc NN

Nf

+= (2.7)

in which, fc is the volume fraction of carbide phase, and Nb, Nc are the numbers of point

counts in binder, and carbide phases, respectively.

ii) Particle size and size distribution

Particle size and size distribution also affect the properties of WC-Co composites.

This includes the Co binder mean free path and the distribution of free paths, as well as

the average size and the size distribution of the carbide particles. Like other engineering

materials, the qualitative effect of particle size on the materials performance is well-

established. Fine-grained grades give rise to high hardness and low fracture toughness,

29

and coarse-grained grades produce low hardness and high toughness. The average carbide

particle size and binder mean free path can be calculated from the following linear-

intercept equations using stereological principles [21].

bccc

c

NNf

L+

=2

2* (2.8)

)1(

1**1 Cf

fL

c

c

−−

×=λ (2.9a)

c

c

ff

L−

×=1

**2λ (2.9b)

where L* is the average carbide particle size, Ncc and Nbc are the average numbers of

intercepts per unit length of test line with traces of carbide/carbide boundary and

binder/carbide interface, λ* is the binder mean free path, and C is the contiguity of

carbide phase. Note that there are two distinct expressions for the binder mean free path;

one contains the contiguity factor in the denominator (Eq. (2.9a)), and the other does not

(Eq. (2.9b)). Only one of these expressions can be correct.

iii) Contiguity of carbide phase

The contiguity (C) of carbide phase can be defined as the ratio of the

carbide/carbide interface area to the total interface area [22, 23]. The degree of contiguity

greatly influences the properties of composites, particularly if the properties of the

constituent phases differ significantly. Contiguity ranges from 0 to nearly 1 as the

distribution of the carbide phase changes from a totally dispersed to a completely

agglomerated structure in the liquid phase. The extent of continuity is primarily

30

determined by the volume fractions. Generally, the contiguity is higher in high carbide

volume fraction grades than in low carbide volume fraction grades. From this, it can be

imagined that the effects of contiguity on the mechanical properties is similar to that of

the carbide volume fraction. As the contiguity increases, hardness increases and fracture

toughness decreases. It has been reported that the degree of contiguity is also affected by

the carbide particle size; increasing carbide particle size is thought to decrease contiguity

[7]. Some authors suggested that the contiguity is also determined by the sintering time,

and that it is independent of grain size [22, 23]. The contiguity of the carbide phase can

be determined using a linear-intercept analysis by the following formula [21],

bccc

cc

NNNC+

=2

2 (2.10)

iv) Angularity of carbide phase

Angularity has been used quantitatively to describe the degree of faceting in the

carbide grains. Therefore, it is thought to be related to the surface energy anisotropy

and/or the anisotropy of growth rates. To our knowledge, the angularity of the carbide

phase has not previously been quantified and related to the properties of WC-Co

composites. This is probably because it is not possible to estimate angularity using linear-

intercept analysis. Non-angular WC-Co composite grades are thought to have undesirable

mechanical properties, although the actual effect of angularity has not been quantified.

31

2.4 Mechanical behavior of WC-Co composites

2.4.1 Elastic stress and stored energy

i) Stress invariant and strain invariant

Stresses (σij) and strains (εij) are 2nd rank tensors (see Eq. 2.11). In the stress and

strain tensors, the first subscript denotes the normal of plane and the second subscript

stands for the direction of forces. Fig. 2.16 schematically illustrates the stress tensor.

, (2.11) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

333231

232221

131211

σσσσσσσσσ

σ ij

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

333231

232221

131211

εεεεεεεεε

ε ij

Fig. 2.16 : Schematic illustration of stress tensor. Only x (x1) and y (x2) components are shown.

x1

x2

σ22

σ21

σ12

σ11 σ11

σ12

σ21

σ22

32

The stress vector at any point, iijj nt σ= , can be transformed to ‘principal stress’

(whose off-diagonal components are all zero) by taking principal stress directions ( ) as

a basis for a new coordinate system.

'in

(2.12) ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡↔

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡==

'''

000000

3

2

1

3

2

1

3

2

1

333231

232221

131211

nnn

nnn

nt iijj σ

σσ

σσσσσσσσσ

σ

where tj is the stress vector, ni is the basis vector, (σ1, σ2, σ3) is principal stress, and is

the basis for a new coordinate system. In seeking these principal stresses, the three stress

invariants can be determined by:

'in

3213

1332212

3211

)(σσσ

σσσσσσσσσ

=++−=

++=

III

(2.13)

where I1, I2, I3 are the three stress invariants. The detailed description for deriving stress

invariants (Eq. (2.13)) can be found elsewhere [24, 25]. In Chapter 5, the stress invariant

1 (I1) distributions are frequently presented for the analysis of the stress state in WC-Co

composites.

Similarly, the strain invariants are given by:

3213

1332212

3211

')('

'

εεεεεεεεε

εεε

=++−=

++=

III

(2.14)

33

where I1’, I2’, I3’ are the three strain invariants, ε1, ε2, and ε3 are the principal strains

along 1, 2, and 3 principal strain directions.

ii) Elastic energy density

The elastically stored strain energy (u, elastic stain energy per unit volume) is

important because it influences the amount of external work that must be done to initiate

failure by a brittle fracture mechanism. This will be described in more detail in Chapter 5.

The stored energy is calculated in the following way:

∑=ji

ijiju,2

1 εσ (i,j=1,2,3) (2.15)

2.4.2 Elastic and plastic behaviors

WC-Co composites are composed of a brittle carbide phase and a ductile binder

phase. Elastic and plastic behaviors can be described by the linear and non-linear

responses to applied stresses. Whereas fracture in ductile materials often involves in the

energy absorption during the plastic deformation, failure in brittle materials occurs at the

elastic regime. The composites of interest here contain both types of materials, and both

failure mechanisms are thought to occur.

i) Elastic constants

In an elastic deformation regime, the elastic strain is usually found to be

proportional to the applied stress, which can be expressed by the simple Hooke’s law

(one dimensional case):

34

εσ E= (2.16)

where σ is the applied stress, ε is the strain, and the proportionality E is Young’s modulus

of elasticity. The generalized form of Hooke’s law accounts for the three dimensional

anisotropic linear response to the applied stresses:

klijklij C εσ = (i,j,k,l=1,2,3) (2.17a)

klijklij S σε = (i,j,k,l=1,2,3) (2.17b)

where σij is the stress tensor (2nd order), εij is the strain tensor (2nd order), Cijkl is the

stiffness tensor (4th order), and Sijkl is the compliance tensor (4th order). Since the

properties are not altered by switching i ↔ j (σij=σji, εij=εji, and Cijkl=Cjikl) and i,j ↔ k,l

(Cijkl = Cklij), nine components of stress and strain tensors will reduce to six components

and 81 components of the stiffness and compliance tensors reduce to 21 independent

variables. The number of independent Cijkl and Sijkl can be reduced by recognizing crystal

symmetries. For example, we need only three and five independent components to

describe the stiffness and compliance tensors for a cubic and hexagonal crystal system,

respectively (see Eq. (2.18)):

(cubic) (2.18a)

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

44

44

44

111212

121112

121211

000000000000000000000000

CC

CCCCCCCCCC

Cij

35

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

=

200000

0000000000000000000

1211

44

44

331313

131112

131211

CCC

CCCCCCCCCC

Cij (hexagonal) (2.18b)

In Eq. (2.18), the conventional abbreviated notation for the stiffness tensor was

used (see Eq. (2.19)): e.g. C1111=C11, C1122=C12,…., C1323=C54,…., etc.

(2.19)

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

↔

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

)6()5()4()3()2()1(

)12()13()23()33()22()11(

ii) Plastic deformation

The plastic deformation behavior of polycrystals is closely related to the slip

processes that occur within individual grains of the polycrystalline. It is believed that

there are more favorable orientations than others for the dislocation glide, which are

usually a crystallographically close packed direction parallel to a closed packed plane.

For example, a (0001) slip plane with a slip direction of >< 0211 is commonly observed

in hexagonal close packed (HCP) materials. However, HCP metals are able to glide on

the prismatic and pyramidal planes as well as the close packed basal planes. Because the

failure model used in this work is based on elastic fracture (see Chapter 5), the

mechanism of the plastic flow and deformation in WC-Co composites will not be

described in detail.

36

2.4.3 Fracture approaches [24-26]

i) Griffith condition

The quantitative relationship between crack size and fracture energy was first

investigated by Griffith. The most important point in the Griffith approach is that it does

not admit plastic deformation: the only work absorbed in the failure is the elastic energy

used to create the fracture surface. Griffith took the stored elastic energy density per unit

volume (elastic energy density) to be E22

1 2σσε = , where E is the elastic modulus

(Young’s modulus) (see Section 2.4.1 ii)). The material volume (per unit thickness)

influenced by the crack advance is 2πc2 (plane stress condition), where 2c is the interior

crack length. The volume energy released during the crack advance is then E

c 22πσ . On

the other hand, the crack surface energy increases by 4cγ, where γ is the surface energy

per unit area. Therefore, at the same stress, any cracks above the critical size, c*, will

advance spontaneously. The critical size, c*, can be determined by the following

condition at a given stress:

0422

=⎥⎦

⎤⎢⎣

⎡+− γπσ c

Ec

dcd (plane stress condition) (2.20)

2

2*πσ

γEc = (plane stress condition) (2.21)

Inversely, the critical stress, σ*, at a given crack size can be determined by:

37

2/12* ⎟

⎠⎞

⎜⎝⎛=

cE

πγσ (plane stress condition) (2.22)

Here, for a given crack length of c, if the stress is greater than σ*, then fracture is

spontaneous.

ii) Energy release rate and critical intensity factor

The energy release rate approach to fracture includes plastic deformation. The

energy release rate, G, is the energy released per unit area of crack advance. In this

approach, if G is greater than the critical energy release rate, GC, then the crack advances

spontaneously. In general, GC is not a material property, but it depends on the geometry

(crack size, 2c) of test specimen and the applied load (P). GC is usually determined by the

following equation:

dcdS

tPGC 4

2

= (2.23)

where t is the thickness of specimen and S is the compliance.

The stress intensity factor, cK πσ= , is generally used as a fracture criterion

(see Section 2.5.1 ii) for details). Combining Eq. (2.22), the critical stress intensity factor,

KC, is proven to be:

)1(

)( 2ν−=

EGmPaK C

C (plane strain condition) (2.24a)

EGmPaK CC =)( (plane stress condition) (2.24b)

38

where ν is the Poisson’s ratio. Here, if the stress intensity factor at a given situation is

greater than KC, fracture occurs.

2.5 Microstructure-property relations of WC-Co composites

2.5.1 Mechanical properties [24-26]

i) Hardness

Hardness is a measure of resistance to penetration. Since the carbide phase has a

relatively hard and brittle character, and the binder phase is soft and ductile, hardness is

primarily accounted for by the carbide phase. In WC-Co composites, therefore, the factor

with the most profound influence on apparent hardness is the volume fraction of the

carbide phase. The second factor for determining hardness is the particle size of the

carbide phase. The schematic relationship between these two microstructural parameters

and hardness is shown in Fig. 2.17(a); high hardness is found when the carbide volume

fraction is high and the carbide grain size is small. All the WC-Co composite grades are

characterized by extremely high hardness values, generally expressed in terms of Vickers

(HV) hardness or Rockwell A (HRA) hardness. The Vickers hardness test uses a square-

base diamond pyramid as an indentor. This test utilizes the surface area of indentation as

a measure of hardness. The unit of Vickers hardness test is [kg/mm2] (the load divided by

the surface area of indentation). Instead of the surface area of indentation, the depth of

indentation (under a constant load) is used in Rockwell hardness test. Since there are

several arbitrary scales for Rockwell hardness depending on the applied load, it is

39

important to specify which scale is being used. In this thesis, the hardness of WC-Co

composites were measured by Vickers hardness tests.

ii) Fracture toughness

Fracture toughness is a measure of the energy required for mechanical failure.

Since the ductile binder phase has the ability to absorb energy via plastic deformation,

fracture toughness generally exhibits an inverse relation to hardness; low carbide volume

fraction and large particle sizes enhance toughness (see Fig. 2.17(b)). The general

procedure for measuring fracture toughness involves notching and pre-cracking the

specimen, applying a load so that the crack grows, and measurement of the crack

advancement. Fracture toughness is not an intrinsic material property. In other words, it

depends not only on the materials themselves, but also on the geometry (size) of the

material being fractured. For this reason, fracture toughness is usually designated as KIC

which is known as the plane strain fracture toughness (or critical stress intensity factor).

KIC is independent of the specimen dimensions (size), assuming the test piece is

sufficiently large. The plane strain fracture toughness can be estimated through the

formula [25]:

cK FIC απσ= (2.25)

where σF is the stress required to propagate a interior crack of length 2c, α is a constant

dependent on the precise crack shape (close to unity). In Eq. (2.25), the plane fracture

toughness (usually called fracture toughness) has the units of [ mPa ].

40

(a) (b) Fig. 2.17 : Schematic plots of the relationship between the mechanical properties and the microstructural parameters. (a) Hardness as a function of the carbide volume fraction and grain size, and (b) fracture toughness as a function of the carbide volume fraction and grain size.

iii) Transverse rupture strength (TRS)

It is customary to perform a tensile test to measure the strength of a material.

However, owing to their extreme hardness, WC-Co composites do not respond well to

the tensile and/or compression tests. Instead of the tensile test, a transverse rupture

strength (TRS) test that bends a specimen until failure is used to measure the fracture

strength of WC-Co composites. A typical TRS test for cemented carbides is illustrated in

Fig. 2.18. A formula for TRS with a rectangular piece is given by [27]:

223whFLTRS = (2.26)

where, F is the applied force to fracture, L is the distance between two parallel supports,

w is the width, and h is the height of test specimen piece (see Fig. 2.18). Because the TRS

test estimates the fracture strength of bending, TRS values are expressed in terms of [Pa],

the same units as strength and/or stress. Despite the common use of the TRS test for

41

measuring fracture strength of bending, the relationship between microstructure and TRS

is not well-established. Sometimes, TRS said to decrease with carbide volume fraction

[28]. It is, however, a matter of controversy. For example, Gurland proposed that a

maximum TRS occurs at intermediate binder content [7]. TRS is also influenced by the

particle size. Although it is relatively well understood that TRS has the inverse relation to

the particle size of carbide, as is the case with hardness, the quantitative relationship is

not clear. As mentioned in the introduction chapter (see Section 1.2), the numerical

approximation of the TRS (equivalently, fracture strength) in WC-Co composites is one

of the primary goals in the current work.

Fig. 2.18 : Schematic illustration of the 3-point transverse rupture strength (TRS) bend test. L is the distance between two parallel supports, b is the with, and h is the height of the test piece.

2.5.2 Past models for microstructure-property relations [29-41]

Since the mechanical properties of WC-Co composites depend on microstructural

features, there have been a number of efforts to develop theoretical and empirical models

for microstructure-property relationships. Note that in all of these cases, the measured

42

microstructural parameters were calculated using stereological linear-intercept analysis.

Also, they can be considered as ‘semi-empirical’ models, since all of them contain

empirical or experimental coefficients based on the measured microstructural factors. In

the remainder of this section, some previous models for hardness and fracture toughness

are briefly reviewed (The models for strength (or, TRS) of the composites were rarely

proposed).

i) Hardness

Most of the past hardness models consist of two parts, a contribution from the

carbide phase and a contribution from the binder phase. Attempts were made to relate the

hardness of WC-Co composites in terms of the hardness of WC and Co. For the in-situ

hardness of WC and Co, the empirical relation according to Hall-Petch is frequently used:

sizegrain

k HHardness

'+= (2.27)

where H’ (hardness of a single crystal) and k (a proportionality factor) are constant.

The most widely accepted hardness model for WC-Co composites is the Lee and

Gurland model [29]. The carbide skeleton hypothesis led them to calculate the continuous

carbide volume fraction (fcC) and they proposed the following equation with the concept

of plastic limit analysis:

C)f( HC f HHV) Hardness ( cbcc −+= 1 (2.28)

*1.231382

LH c += ,

*7.12304

1λ+=bH (kg/mm2) (2.29)

43

where and are the in-situ hardness of the carbide and binder phase, respectively,

is carbide volume fraction, C is contiguity, is stereologically measured linear

intercept WC particle size [µm] (Eq. (2.6)), and

cH bH

cf *L

*1λ is stereologically calculated binder

mean free path [µm] (Eq. (2.7a)). On right-hand side of Eq. (2.28), the first term

describes the effect of WC hardness and second term describes the contribution from Co

to the total hardness of composites. Here, the constants in Eq. (2.29) were calculated

empirically based on 26 different specimen grades.

Another recent model is the Enqvist et al. model [31]. They rejected the carbide

skeleton hypothesis and adopted a model in which hardness decays exponentially with

the size of the binder mean free path, *2λ . They suggested the following equation that

includes only the carbide particle size ( ) and volume fraction ( ) parameter, and

avoids computing contiguity (C). Note that λ

*L cf

2* (which is incorrect) is used, instead of λ1*.

bk

bc H eHH HV) Hardness ( +−=−

*2

)(λ

(2.30)

* 1.2

2680693L

H c+

+= , 825=bH (kg/mm2) (2.31)

where is the empirically measured hardening range factor (0.35 µm). k

ii) Fracture toughness

In most of the previous models, the authors tried to find a proper expression for

the strain energy release rate (G). As described before (see Section 2.4.3 ii)), the strain

energy release rate (G) can be defined as the rate at which energy is transferred from the

44

elastic stress field of a crack to the plastic process of crack extension. The critical value

of G that makes the crack propagate to fracture, GIC, is the critical strain energy release

rate (plane strain condition), and it is linked to the critical stress intensity factor (KIC,

plain strain fracture toughness) by the following formula:

)1(

)( 2ν−=

EGmPaK IC

IC (2.32)

where E is Young’s modulus, and ν is Poisson’s ratio.

Pickens and Gurland [32] suggested that the critical strain energy release rate

( ) is linearly related tothe binder mean free path (ICG λ ) and yield strength (σy):

yICG αλσ≅ (2.33)

where α is the linear coefficient and yσ is the in-situ yield strength. Pickens and

Gurland predicted the in-situ yield strength based on the Lee and Gurland hardness model

[29]. In this model, they also assumed a linear relationship between hardness and yield

strength ( yH σ3≅ ).

Sigl and Fischmeister [37] presented a more elaborate model for the prediction of

fracture toughness. They defined four possible crack paths (carbide, binder,

carbide/carbide, and binder/carbide), and modeled the strain energy release rates of these

four paths.

')()( ICcccfbcbcbbIC GAAArArG +++= σ (2.34)

45

12 +

−+=

c

ysyf φ

σσσσ (2.35)

where is the area fractions of crack path through the binder (transgranular),

binder/carbide interfaces (intergranular), the carbide (transgranular), carbide/carbide

interfaces (intergranular), respectively, and are the mean sizes of plastic zone of

the binder and binder/carbide interfaces (0.05 µm), respectively,

cccbcb AAAA ,,,

br bcr

fσ is the mean flow

stress of binder, sσ is the saturation stress (12.3 GPa) of binder, yσ is the yield strength

of binder, cφ is the characteristic strain (0.63), and is the fracture energy of carbide

and carbide/carbide boundaries (50 J/m

'ICG

2). Empirical relations were used to calculate the

area fractions, [36]. cccbcb AAAA ,,,

More recently, Ravichandran [40] suggested that the strain energy release rate

could be obtained by the sum of the fracture energy of carbide phase ( ) and

fracture resistance of binder phase (

'ICcGf

λχσ ocf )1( − ).

λχσ ocICcIC fGfG )1(' −+= (2.36)

c

ccIC E

KG

)1('

2ν−= (2.37)

βσσ

χo

eff= , )]1

(3

1[c

c

o

eff

ffk

−+=

σσ

(2.38)

where is the fracture energy of carbide phase, 'ICG oσ is the bulk flow stress of binder

(850 MPa), and χ is the rupture parameter of binder, ccc EK ,,ν are the fracture

toughness, Poisson’s ratio, and modulus of carbide, respectively, effσ is the effective

46

flow stress of binder, β is the ratio of the critical crack tip to binder thickness (2.0), and

is the maximum shear factor (0.577). k

We tested the performances of those previous models using the current specimen

grades studied in this thesis. It turned out that the calculated mechanical properties based

on the previous models do not match up well with the measured data. This will be

presented in Section 3.4.1 and 3.4.2.

2.6 Numerical models for structural analysis

The mechanical response of a material can be modeled by both analytical and

numerical methods. Although analytical methods provide closed formed exact solutions,

they are limited to specific cases; analytical methods cannot be rigorously applied to

complicated microstructures such as those considered in this work. For this reason, since

the 1980’s, with the increasing capabilities of computers, attempts have been made to

predict stress-strain distributions in microstructures using computer-based numerical

simulations. Among diverse numerical analysis methods, the finite element method (FEM)

is accepted as a powerful tool to approximate the solutions of differential equations

governing various physical phenomena. Past attempts to calculate mechanical properties

of WC-Co composites using FEM focused only on the prediction of the crack

propagation paths using stress-strain analysis within relatively small and localized areas

[42, 43].

47

2.6.1 Finite element methods (FEM) for stress-strain calculations

Like in many other numerical analysis tools, discretizing the domain is the first

step in the FEM. The domain is discretized into a finite number of ‘elements’ with

polygonal shapes that represent the properties of a corresponding local area. Due to its

simplicity, the triangle is the most frequently used element type in two-dimensional finite

element analysis. The key to the FEM is utilizing the ‘principle of minimum potential

energy’ in these discretized finite elements. The principle of minimum potential energy

requires that, when a system reaches an equilibrium state, the first variation of the total

potential energy of the system must be zero to minimize the total potential energy. Thus,

if we take Π, UI, and W as the total potential energy, internal potential energy, and work

done to the system, respectively, the principle of minimum potential energy can be

written in the following way:

(2.39) WU I −=Π

(2.40) WUWU II δδδδΠδ ==−= ,0

Here, in the stress-strain analysis, the variations of the internal energy and work

done to the system are defined as:

(2.41) ∫=

Ω

Ωδεσδ dU ijijI )(

(2.42) ∑∫∫=

++=M

mi

miiiii updusdubW

1)()()( δΓδΩδδ

ΓΩ

in which σij is the stress tensor, εij is the strain tensor, ui is the displacement, bi is the

body force, si is the surface tension, and pim is the mechanical point load at node m. Here,

48

Ω and Γ represent the respective volume and boundary of the domain, and M denotes the

total number of nodes. The preceding equation (Eq. (2.42)) is applicable only to the

system under mechanical loads. For a system under thermal loads, the work variation

term (Eq. (2.42)) should be modified. From Eq. (2.41) and Eq. (2.42), the principle of

minimum potential energy reduces to:

(2.43) ∑∫∫∫=

++=M

mi

miiiiiijij updusdubd

1)()()()( δΓδΩδΩδεσ

ΓΩΩ

In Eq.2.43, the only unknown variable is the displacement (ui), because the elastic

relations of strain-displacement and stress-strain are well-established:

kijkij uL=ε (2.44)

klijklij C εσ = (2.45)

where Lijk is the linear differential operator tensor, and Cijkl is the stiffness tensor. Since

Eq. (2.43) is analytically unsolvable in a complex structure, a basis function (shape

function) is introduced in order to numerically resolve the problem. Employing a basis

function also allows us to calculate the displacements at all points inside an element in

terms of nodal values (displacements of nodes). A general finite element formulation for

the definition of a basis function is expressed by:

(2.46) ∑=

=≈M

m

mi

mii uNuu

1

ˆ

where Nm is the basis function at node m and ui

m is the displacement vector whose

components are the approximations of the displacements at node m. The basis function

49

(Nm) varies linearly from 1 at node m to 0 at the remaining nodes. Then, the strains and

stresses at any point follow as:

, where (2.47) ∑∑==

===M

m

mi

mijk

M

m

mk

mijkkijkij uBuNLuL

11ε m

ijkmijk NLB =

(2.48) ∑=

==M

m

mm

mklmijklklijklij uBCC

1''εσ

Substituting Eq. (2.46)~(2.48) into the minimum potential energy equation (Eq.

(2.43)), the equation becomes (in a matrix form):

(2.49) ∑∫∫∑ ∫==

++=⎥⎦

⎤⎢⎣

⎡ M

m

M

mddd

11)( NpNsNbuCBB T

ΓΩΩ

ΓΩΩ

This equation set can be expressed in terms of the standard algebraic equation system:

fK =φ (2.50)

where

, ∑ ∫=

⎥⎦

⎤⎢⎣

⎡=

M

md

1)(

Ω

ΩCBBK T u=φ , (2.51) ∑∫∫=

++=M

mdd

1NpNsNbf

ΓΩ

ΓΩ

K is called the global matrix (or, stiffness matrix) which can be obtained by

summing the contributions from the individual element matrices whose components are

defined by:

(2.52) ∫=

e

dΩ

Ωme

Tle

lme CBBK )(

50

Likewise, f (which is called the load vector) can be decomposed into a set of

element load vectors:

(2.53) pNsNbNf llll

e ++= ∫∫ee

ddΓΩ

ΓΩ

From Eq. (2.49) and Eq. (2.50), the solutions of the displacements of individual

nodes are approximated, and also elastic stress-strain distributions can be calculated by

strain-displacement and stress-strain relations (Eq. (2.44) and (2.45)). The detailed

descriptions for the residual stress-strain calculations using FEM can be found elsewhere

[44-47].

2.6.2 Previous analyses for the prediction of crack paths in WC-Co composites

using FEM

In most of the cases, the prediction of crack paths in WC-Co composites has been

a main issue in the past attempts for stress-strain field analysis using FEM. Fischmeister

et al. [42] calculated the distribution of plastic strains and measured the size of plastic

zones in the binder phase. They found that the plastic zone size is smaller than the size of

the binder mean free path and that it depends on the shape of the binder regions, which is

in agreement with experimental observations. In the initial setting for the FEM

calculations, intergranular cracks were assumed to have already formed in the carbide

phase. In other words, only the stress-strain states in the binder phase are analyzed in

their model.

Spiegler and Fischmeister [43] used a FEM analysis to decide whether cracks

propagate along the binder/carbide interface (intergranular) or through the binder phases

51

(transgranular). They explained that the crack path is determined by the angle at which

the crack enters the binder region. It was found that the critical transition angle from

transgranular to intergranular fracture is 25°. Both of the preceding analyses were based

on the average properties of individual phases (Young’s modulus: EWC, ECo, EComposites=

714, 211, 595 GPa, and Poisson’s ratio: νWC, νCo, νComposites=0.19, 0.31, 0.22,

respectively). Another shortcoming of the prior studies is that the calculated regions are

limited to localized areas (small area simulations), including less than 20 carbide grains.

2.6.3 Object-oriented finite element (OOF) model

Recently, a comprehensive package for mechanical property calculations has been

developed, and continues to be improved by NIST (National Institute of Standards and

Technology, Gaithersburg, MD) [47]. The name of this code, object-oriented finite

element (OOF) analysis, derives from the object-oriented programming language in

which it is encoded (in this case, C++).

Based on the FEM, OOF is designed to approximate solutions in stress-strain

states of complex microstructures (including a large number of grains) under mechanical

and/or thermal loads. Its ability to model a large number of grains is the primary merit of

the OOF code. The stress-strain fields of complicated microstructures having more than

several hundred grains can be calculated. It is an obvious advantage over the past analysis

models in which the calculations are confined to small, localized areas. Another

advantage of the OOF package is that the specific materials properties (crystallographic

orientations, stiffness, thermal expansion coefficients, and surface energies of fracture

surfaces) of individual crystals are easily incorporated. In other words, the anisotropic

52

characteristics of each material can be accounted for in the calculation of the internal

stress and strain. In the previous FEM models of WC-Co composites, these

crystallographic anisotropies have not been properly modeled [40, 41]. For example, the

bulk Young’s modulus and Poisson’s ratio were used instead of the stiffness (C11, C33,

C44, C12, and C13) of each phase. OOF also provides a way to simulate crack propagation.

It assumes an element cracks when the supplied stored energy is greater than the surface

energy the element. It is closely associated with ‘brittle fracture’, or ‘Griffith-like

fracture’ (see section 2.4.3 i)). This brittle fracture criterion can be written as:

AL ijijεσγ212 ≤∆ (2.54)

where γ is the surface energy of the cracked interface, ∆L is the edge length, A is the area,

σij is the stress, and εij is the strain of a element, respectively. A typical procedure for

OOF simulation is as follows:

1) Start with a micrograph. The micrograph could be a virtual image from a computer

simulation, or an experimental map from certain microscopes. Any digitalized

microstructure can be used as input for OOF simulations.

2) Select features in the image and assign materials properties. In the OOF code,

individual crystals are usually differentiated by color contrast differences. In each distinct

crystal, corresponding materials properties are then applied (e.g. orientations, elastic

constants, thermal expansion coefficients, and surface energies).

53

3) Perform a virtual experiment with relevant boundary conditions. The simulation is able

to be carried out under mechanical loads (tensile or compression) and/or thermal loads

(quenching or annealing).

4) Calculate and visualize the internal stress and strain. Using a finite element algorithm,

the solutions of stress-strain states are approximated. In the OOF package, these solutions

can be visualized in terms of various stress, strain, or stored energy distributions such as

σxx ,σyy, σzz, σxy ,σyz, σzx, stress invariant 1 (σI, σI=σxx+σyy+σzz), εxx ,εyy, εzz, εxy ,εyz, εzx,

strain invariant 1 (εI, εI=εxx+εyy+εzz), or elastic energy density.

5) Compute the macroscopic mechanical properties. From the calculation of internal

stress and strain, the effective mechanical properties (macroscopic modulus, average

stress, average strain, etc.) can be estimated.

There are some reports on mechanical property calculations using the OOF code

described in this section. Vedula et al. calculated the stresses and elastic energy densities

in polycrystalline alumina. They also compared the stress distributions in orientation

textured and non-textured samples and verified that the textured microstructure exhibits a

low distribution of stress and stored energy [48]. In this document, this texture effect on

the strength of WC-Co composites will be presented in Chapter 5 (see Section 5.3.5).

Chawla et al. used the OOF package for the calculations of Young’s moduli and thermal

expansion coefficients of SiC-Al and WC-Co (double carbide) composites [49]. The

authors demonstrated that the calculations are consistent with the experimental results. In

this literature [49], the detailed description of OOF code was also presented.

54

2.7 Summary

In the preceding sections in this chapter, background theories and models which

are relevant to the thesis have been reviewed. Previously, the WC habits in the WC/Co

interface and the skeletons of WC/WC grain boundaries have been measured in a

qualitative way. All of the measurements of microstructural features were accomplished

by the linear-intercept method. The mechanical properties (especially for the hardness

and fracture toughness) of the composites were modeled as a function of the carbide

volume fraction (fc) and the particle size (average carbide size and mean binder free path).

In the following chapters, we will describe in detail about experimental and analytical

methods for the microstructural characterization from direct measurements, interface

character distributions using a stereological method, and fracture strength calculations

based on finite element simulations.

2.8 References

[1] P.W. Tasker, “The Stability of Ionic Crystal Surfaces”, J. Phys. C, 12, 4977 (1979)

[2] A.D. Krawitz, D.G. Reichel, and R. Hitterman, “Thermal Expansion of Tungsten

Carbide at Low Temperature”, J. Am. Ceram. Soc., 72, No.3, 515 (1989)

[3] R.R. Reeber and K. Wang, “Thermophysical Properties of a-Tungsten Carbide”, J.

Am. Ceram. Soc., 82, No. 1, 129 (1999)

[4] H.E. Exner, “Physical and Chemical Nature of Cemented Carbides”, Int. Metal. Rev.,

No. 4, 149 (1979)

55

[5] L. Ramqvist, “Wetting of Metallic Carbides by Liquid Copper, Nickel, Cobalt and

Iron”, Int. J. Powder Metall., 1, No. 4, 2 (1965)

[6] D.N. French, “The Microstructure and Microproperties of WC-Co Alloys”, Int. J.

Powder Metall., 5, No. 3, 47 (1969)


Mater. Rev., 33, No. 3, 151 (1988)

[8] A.V. Shatov, S.A. Firstov, nad I.V. Shatova, “The Shape of WC Crystals in

Cemented Carbides”, Mater. Sci. Eng. A, 242, 7 (1998)

[9] H.-R. Lee, D.-J. Kim, N.-M. Hwang, and D.-Y. Kim, “Role of Vanadium Carbide

Additive during Sintering of WC-Co: Mechanism of Grain Growth Inhibition”, J. Am.

Ceram. Soc., 86, No. 1, 152 (2003)

[10] S. Kim, S.-H. Han, J.-K. Park, and H.-E. Kim, “Variation of WC Grain Shape with

Carbon Content in the WC-Co Alloys during Liquid-phase Sintering”, Scripta Mater., 48,

635 (2003)

[11] V. Randle and L. Engler, Introduction to Texture Analysis, 1st ed., Taylor & Francis

Ltd, 2002

[12] A. Heinz and P. Neumann, “Representation of Orientation and Disorientation Data

for Cubic, Hexagonal, Tetragonal, and Orthorhombic Crystals”, Acta Crystal., A44, 326

(1988)

[13] A. Morawiec and D.P. Field, “Rodrigues Parameterization for Orientation and

Misorientation Distributions”, Phil.Mag.A, 73, No.4, 1113 (1996)

[14] G.S. Rohrer, Structure and Bonding in Crystalline Materials, 1st ed., Cambridge

University Press, Inc., 2001

56

[15] S. Bartolucci and H.H. Schlosin, “Plastic Deformation Preceding Fracture in

Tungsten Carbide-Cobalt Alloys”, Acta Metall. Mater., 14, No. 3, 337 (1966)

[16] J. Gurland and J.T. Norton, “Role of the Binder Phase in the Cemented Carbide-

Cobalt Alloys”, Trans. Metall. Soc. AIME, 194, 1051 (1952)

[17] A. Henjered, M. Hellsing, H.-O. Andren, and H. Norden, “Quantitative

Microanalysis of Carbide/carbide Interfaces in WC-Co-base Cemented Carbides”, Mater.

Sci. Technol., 2, 847 (1986)


Phys. Stat. Sol. A, 61, 97 (1980)

[19] R. Deshmukh and J. Gurland, “Orientation of Contiguous Tungsten carbide Crystals

in Sintered WC-Co as Determined from Slip Line Traces”, Metallography, 15, 383 (1982)

[20] B. Roebuck and E.A. Almond, “Deformation and Fracture Processes and the

Physical Metallurgy of WC-Co Hardmetals”, Int. Mater. Rev., 33, No. 2, 90 (1988)

[21] E. Underwood, Quantitative Stereology, Addison-Wesly, Inc., New York, 1970

[22] J. Gurland, “An Estimate of Contact and Contiguity of Dispersions in Opaque

Samples”, Trans. Metall. Soc. AIME, 236, 642 (1966)

[23] R.M. German, “The Contiguity of Liquid Phase Sintered Microstructures”, Metall.

Trans. A, 16A, 1247 (1985)

[24] G.E. Dieter, Mechanical Metallurgy, McGraw-Hill, Inc., 1988

[25] T.H. Courtney, Mechanical Behavior of Materials, 2nd ed., McGraw-Hill, Inc., 2000

[26] R.E. Reed-Hill and R.A. Abbaschian, Physical Metallurgy Principles, 3rd ed., PWS

Publishing Co., 1994

[27] Work Instructions for TRS test, Kennametal, Inc., 1998

57

[28] G. Schneider, Jr., Principles of Tungsten Carbide Engineering, 2nd ed., Society of

Carbide and Tool Engineers, 1989

[29] H.C. Lee and J. Gurland, “Hardness and Deformation of Cemented Tungsten

Carbide”, Mater. Sci. Eng., 33, 125 (1978)

[30] M.T. Laugier, “A Microstructural Model for Hardness in WC-Co Composites”,

Acta Metall., 33, No. 11, 2093 (1985)

[31] H. Engqvist, S. Jacobson, and N. Axen, “A Model for Hardness of Cemented

Carbides”, Wear, 252, 384 (2002)

[32] J.R. Pickens and J. Gurland, “The Fracture Toughness of WC-Co Alloys Measured

on Single-edge Notched Beam Specimens Precracked by Electron Discharge Machining”,

Mater. Sci. Eng., 33, 135 (1978)

[33] M. Nakamura and J. Gurland, “The Fracture Toughness of WC-Co Two-Phase

Alloys – A Preliminary Model”, Metall. Trans. A, 11A, 141 (1980)

[34] V.D. Krstic and M. Komac, “Toughening in WC-Co Composites”, Philo. Mag. A,

51, No. 2, 1985 (191)

[35] D.K. Shetty, I.G. Wright, P.N. Mincer, and A.H. Clauer, “Indentation Fracture of

WC-Co Cermets”, J. Mater. Sci., 20, 1873 (1985)

[36] L.S. Sigl and H.E. Exner, “Experimental Study of Mechanics of Fracture in WC-Co

Alloys”, Metall. Trans. A, 18A, 1299 (1987)

[37] L.S. Sigl and H.F. Fischmeister, “On the Fracture Toughness of Cemented

Carbides”, Acta Metall., 36, No. 4, 887 (1988)

[38] D. Han and J.J. Mecholsky, “Fracture Analysis of Cobalt-bonded Tungsten Carbide

Composites”, J. Mater. Sci., 25, 4949 (1990)

58

[39] D.B. Marshall, W.L. Morris, B.N. Cox, and M.S. Dadkhah, “Toughening

Mechanisms in Cemented Carbides”, J. Am. Ceram. Soc., 73, No. 10, 2938 (1990)

[40] K.S. Ravichandran, “Fracture Toughness of Two Phase WC-Co Cermets”, Acta

Metall. Mater., 42, No. 1, 143 (1994)

[41] J.M. Densley and J.P. Hirth, “Fracture Toughness of a Nanoscale WC-Co Tool

Steel”, Scripta Mater., 38, No. 2, 239 (1998)

[42] H.F. Fischmeister, S. Schmauder and L.S. Sigl, “Finite Element Modeling of Crack

Propagation in WC-Co hard Metals”, Mater. Sci. and Eng. A, 105/106, 305 (1988)

[43] R. Spiegler and H.F. Fischmeister, “Prediction of crack paths in WC-Co alloys”,

Acta. Metall. Mater., 40, No. 7, 1653 (1992)

[44] O.C. Zienkiewicz and K. Morgan, Finite Elements and Approximation, John Wiley

& Sons, Inc., 1983

[45] J.N. Reddy, An Introduction to the Finite Element Method, 2nd ed., McGraw-Hill,

Inc., 1993

[46] E.B. Becker, G.F. Carey, and J.T. Oden, Finite Elements – An Introduction :

Volume 1, Prentice-Hall, Inc., New Jersey, 1981

[47] W.C. Carter, S.A. Langer and E.R. Fuller Jr., The OOF Manual (version 1.0.8.6),

National Institute of Standards and Technology (NIST), (2001); also available online at

http://www.ctcms.nist.gov/oof

[48] V.R. Vedula, S.J. Glass, D.M. Saylor, G.S. Rohrer, W.C. Carter, S.A. Langer, E.R.

Fuller, “Residual-stress Predictions in Polycrystalline Alumina”, J. Am. Ceram. Soc., 84,

No. 12, 2947 (2001)

59


[49] N. Chawla, B.V. Patel, M. Koopman, K.K. Chawla, R. Saha, B.R. Patterson, E.R.

Fuller, and S.A. Langer, “Microstructure-based Simulation of Thermomechanical

Behavior of Composite Materials by Object-oriented Finite Element Analysis”, Mater.

Character., 49, 395 (2003)

60

Chapter 3

Microstructural Characterization

This chapter describes the methods used to characterize the microstructures of

WC-Co composites and presents the results of the analysis. Atomic force microscopy

(AFM) and orientation imaging microscopy (OIM) have been used to extract the

binder/carbide and carbide/carbide interface positions in WC-Co composites with carbide

volume fractions from 0.7 to 0.9 and mean carbide grain diameters from one to seven

microns. In this work, we measure the microstructural features using direct measurements

of two-dimensional features rather than the conventional linear-intercept method. We will

start this chapter with descriptions of the sample preparation method and direct

measurement technique used throughout this thesis. After presenting the quantitative

results, the relationship between microstructures and properties will be discussed.

3.1 Samples

Nine specimen grades are used for this thesis (samples were prepared at

Kennametal Inc., Latrobe, PA, USA). The first eight, labeled A-H, contain no intentional

alloying additions (straight grades). The last straight specimen grade (grade H) is known

as a non-angular grade which, in contrast to the grades A through G, is comprised of

carbide grains with less polygonal shapes. The ninth sample, specimen grade I, is an

alloyed grade that contains Ta, Ti, and Nb at concentrations of 4.7, 2.1, and 0.5 weight

61

percent (w/o), respectively. This sample is also referred to as a plate-like grade and it

contains carbides that are elongated in directions perpendicular to the [0001] axis.

Throughout this thesis, the same notation for the nine specimen grades (A~I) will be used.

The phase fractions for each specimen grade are listed in Table 3.1. These commercially

produced samples were consolidated via liquid phase sintering for approximately 45 min.

at temperatures in the range of 1400 °C to 1600 °C. Materials with the highest binder

fractions were sintered at the lower end of this temperature range and those with the least

binder were sintered at the higher temperature. Images from these samples using atomic

force microscopy (AFM) are shown in Fig. 3.1 (same scale).

Table 3.1 : Carbide volume fractions for each specimen grade.

62

A

H B C

D E

20 µmF G I

I

Fig. 3.1 : Images of nine specimen grades (A~I) from atomic force microscopy (AFM). For comparison, same scale is used for upper nine grades. In the bottom, AFM image (different scale,

14×14 µm2) of grade I (plate-like grade) is re-shown for a clear visualization.

63

After polishing using a diamond abrasive, the samples were etched in Murakami’s

reagent (10 g potassium ferricyanide+10 g sodium hydroxide+100 ml distilled water) for

1 min. The reagent preferentially attacks the carbide phase with different rates and

produces topographic differences at the binder/carbide interfaces and the carbide/carbide

boundaries that lead to sharp contrast in atomic force microscopy (AFM) images (see Fig.

3.1 and 3.2). Here, dark contrast corresponds to the topologically lower surface (the

binder phase before the etch and the carbide phase after the etch), and white contrast

represents the relatively higher surfaces (the carbide phase before the etch and the binder

phase after the etch).

(a) (b) Fig. 3.2 : Advantage of etching in Murakami’s reagent (from grade A, 60×60 µm2). Same sections are shown. (a) Before etching, and (b) after etching in Murakami’s reagent for 1 min. White contrast corresponds to the topologically higher region and dark contrast corresponds to lower region.

64

3.2 Microstructure and property characterization methods

3.2.1 Atomic force microscopy (AFM)

A Park Scientific Instruments Cp or a Thermomicroscopes M5 was used for all

AFM imaging. Gold-coated, sharpened pyramidal Si3N4 probes (Thermomicroscopes

ML06A-F) were used to obtain contact images. The field of view for each AFM image

was selected based on a rough estimate of WC grain size (80×80 µm2 for H, 60×60 µm2

for A, 40×40 µm2 for F and G, 20×20 µm2 for B, C, D, and E, and 14×14 µm2 for I) such

that each AFM section contains 100~180 carbide grains. The microscope was

programmed to automatically record 100 images in a 10 by 10 array such that there was

approximately a 10 % overlap with adjacent images in the array. Thus, the total scanned

areas ranged from 710×710 µm2 for sample H to 122×122 µm2 for sample I. This

imaging procedure allows the shape of many grains to be determined with high resolution.

Since there are 512×512 pixels in each of the AFM images, the pixel spacing is 0.16 µm

(H), 0.12 µm (A), 0.08 µm (F and G), 0.04 µm (B, C, D, and E) and 0.03 µm (I). Typical

AFM digital images (after etching in Murakami’s reagent) of WC-Co composites are

illustrated in Fig. 3.3(a) (from grade A). The Individual WC grains are dissolved at

different rates by the etching solution and, therefore, have various shades of darker

contrast. The differential etching allows both WC/Co and WC/WC boundaries to be

easily identified. In our experience, the vertical sensitivity of the AFM allows the phase

boundaries in WC-Co composites to be resolved more clearly than optical microscopy

(OM) or scanning electron microscopy (SEM). This is particularly important for

65

measuring the shapes of the crystals with sub-micron diameters that are found in these

materials.

3.2.2 Orientation imaging microscopy (OIM)

In this work, carbide grain orientations were determined using an OIM system

(TexSEM Laboratories, Inc.) incorporated in a Phillips XL40 FEG scanning electron

microscope (SEM). These measurements also made it possible to distinguish ambiguous

carbide/carbide and cobalt/carbide phase boundaries. For each sample, carbide

orientations were mapped in the same regions imaged by AFM. Orientation

measurements were made at intervals of 1.2 µm (H), 1.0 µm (A), 0.4 µm (F and G), 0.2

µm (B, C, D, and E) and 0.15 µm (I) over total areas of 800×800 µm2, 600×600 µm2,

360×360 µm2, 180×180 µm2 and 150×150 µm2, respectively. All OIM images were

recorded at a specimen tilt of 70°, an accelerating voltage of 20 kV, and a spot size of 5.

A typical OIM inverse pole figure (IPF) map is shown in Fig. 3.3(b) (from grade A, same

section as Fig. 3.3 (a)). Here, the color contrast stands for the normal direction of the

surface plane in each WC crystal.

During these studies, the orientations of Co grains were not recorded. Several

factors make it difficult to collect reliable electron backscattered patterns (EBSPs) from

the Co phase. First, because of the small size of the binder pools and the topographic

differences between carbide and binder phases, the patterns can be affected by

superposition and shadowing. Second, because of the different mechanical properties of

the two phases, the conditions required for polishing the composites to yield good EBSPs

from the carbide grains created so much plastic damage in the binder that diffraction from

66

the Co was not observed. Polishing procedures that produced acceptable surfaces of pure

Co samples were not aggressive enough to polish the carbide phase. The only way we

found to prepare samples so that EBSPs could be simultaneously recorded from the

carbide and the binder was to first polish the samples to prepare the carbide phase, and

then anneal the sample at 800 °C (in a reducing environment) to heal some of the plastic

damage in the binder phase. It was then possible to record EBSPs from the largest binder

pools. Limited studies of the sample with the highest volume fraction of Co indicated

that neighboring pools of Co had the same orientation and that the Co grain size was

larger than the carbide grain size.

3.2.3 Microstructure analysis

The positions of binder/carbide interfaces and carbide/carbide boundaries in AFM

images are digitized using a program that automatically records the vector components of

binder/carbide (lbc) and carbide/carbide (lcc) line segments when the user traces them with

a computer mouse. Hence, polygonal closed contours of WC particles consisting of

binder/carbide and carbide/carbide line segments joined at N vertices can be constructed.

For each of the nine specimen grades, the boundaries in 20 AFM images were

skeletonized so that for each grade, the geometric results are based on measurements of

approximately 2500 carbide grains. An example of a skeletonized map is shown in Fig.

3.4. The grain area (A) of carbide is then computed as the total number of pixels within a

closed polygon, multiplied by the area per pixel. Grain diameters (d) are computed by

assuming that the grain area is circular (d=2(A/π)1/2). The contiguity of carbide phase (C)

is computed according to the following expression:

67

(a)

(b) OIM in the same area (from grade A). (a) 5×4 array of

AFM images, (b) OIM inverse pole figure (IPF) map (Co phase is not indexed). Fig. 3.3 : Typical images of AFM and

68

∑

∑+

= jbc

icc

iicc

lll

C)2(

2

ji,

(3.1)

By tracing the carbide grains and assigning the line segments as binder/carbide or

arbide

4

L

c /carbide interfaces, the number of vertices (N), the grain area (A), the grain

diameter (d), and the contiguity (C) are determined by direct observation. For comparison,

however, we also examined quantities determined from linear-intercept lengths and

quantities computed using stereological equations based on the frequency of linear-

intercepts. We use all 512 lines in each of the 20 skeletonized AFM image as test lines,

so that approximately 2 (x- and y-direction) ×10 test lines are used for each specimen

grade. From these test lines, we determined the mean carbide intercept length, L, and the

mean binder intercept length, λ, and the average number of intercepts per unit length of

test line with carbide/carbide boundaries (Ncc) and binder/carbide interfaces (Nbc). The

stereologically determined carbide grain size, *, is:

bccc

c

NNf2

L+

=2

* (3.2)

where fc is the volume fraction of the carbide phase. The stereologically derived

contiguity (C*) is:

bccc

cc

NNN

C+

=2

2* (3.3)

In previous work, there are two different expressions by which the stereologically

etermined binder mean free path (λ*) is computed:

d

69

)1(

**1 CfL

c

c

−)1( f−

=λ (3.4a)

c

c

ff

L)1(

**2−

=λ (3.4b)

[1-3] and λ2* w

see that λ1* turns out to be the correct expression for WC-Co composites.

where λ1* was used in references as used in references [4, 5]. We shall

(a) (b) Fig. 3.4 : Example of direct measurements (from grade D, 40×40 µm2). (a) Original AFM image,

skeletonized map.

s of mechanical properties

All of the measured mechanical properties are summarized in Table 3.2. These

at Kennametal Incorporated (Latrobe,

PA, USA). The values in parentheses are the standard deviations in each grade.

(b)

3.2.4 Measurement

measurements were all conducted by collaborators

70

Table 3.2 : Mechanical properties for each grade. The values in the parenthesis denote standard deviations. - notation means N/A (not available).

for the mean hardness. All the measured

rdness values are at the vicinity of mean hardness, and the standard deviations (grades

A-H) of

d 45

mm lon

5×5×20 mm3. More than 15 independent measurements were averaged, and the standard

Hardness was measured on a Vickers hardness scale (HV, 30-kg load). Five

measurements of each grade were averaged

ha

each grade range from 2 to 10 kg/mm2, which is on the order of 1 % or less.

To measure the fracture toughness, a 10000 kg load cell was used for pre-cracking

of test specimens and a 230 kg load cell was used for breaking specimens in the four-

point bending failure test. The specimen dimensions were 6 mm high, 3 mm wide, an

g. For statistically reliable information, 20 measurements for each specimen grade

were performed. The standard deviations in the Table 3.1 (for grades A-D, F, and G) fall

in the range of 0.29~0.45 MPa, which are in the range of 2 to 3 %.

The dimensions for the transverse rupture strength (TRS) test specimens were

71

deviation for three grades (grades A, B, and F) ranges from 0.05 to 0.13 GPa, which

range from 3 to 7 %.

3.3 Results

Table 3.3 : Summary of the results of microstructures. d : carbide grain diameter by direct measurements, L : carbide grain size by direct measurements, L* : carbide grain size by stereology (linear-intercept), l : binder mean free path by direct measurements, l* : binder mean free path by stereology, f’c : theoretical carbide volume fraction, fc : measured carbide volume fraction by direct measurements, C : contiguity by direct measurements, N : ave. number of vertices per grain

72

The measured microstructural parameters are summarized in Table 3.3. The

values in parentheses denote standard deviations in the measurements.

3.3.1 Grain size, binder mean free path, and size distributions

The average grain diameter and the average intercept length are comparable. The

ratios of

lly derived mean

intercept length is comparable to the measured value. The assumption of the circular

rger values. Random

intercep

c

the two values fall in the range of 0.99 to 1.13, such that the intercept length, L,

is usually smaller than the effective grain diameter, d. The stereologica

section plane in the calculation of d leads to the systematically la

t lengths through the polygonal carbide grains are, on average, smaller than

through a circle of the same area. Note that the volume faction of the carbide phase (f ) is

systematically lower than the value calculated on the basis of the starting materials. This

is expected, since some of the carbide will dissolve into the Co during sintering. The

amount of carbide lost to dissolution varies from 0.5 to 5.6 %, which is less than the

equilibrium value of about 10 % at the 1400 °C WC-Co eutectic.

The distributions of grain sizes were found to be surprisingly uniform. When the

size distributions are normalized, all exhibit uni-modal and that log-normal size

distributions are close to the Gaussian distributions (see Fig. 3.5(a) and Fig. 3.6). For

comparison, the Gaussian, Hillert [6, 8], and Louat [7, 8] distributions are also shown. A

Gaussian distribution is often called a ‘normal distribution’ which is given by:

⎥⎦

⎤⎢⎣

⎡ −−=

2)(exp1)( µRRf (3.5) 222 σπσ

73

where f(R) is the probability function, R is the variate (grain diameter or radius), µ is the

mean of the variate, and σ is the variance (standard deviation) of the variate. The so-

alled ‘standard normal distribution’ can be expressed by taking µ=0, and σ=1:

c

⎥⎦

⎤⎢⎣

⎡−=

2exp

21)(

2RRfπ

(3.6)

mpletely

mbedded in an averaged environment. In the mean field theory, the overall flux (j) of

of the diffusion term an

Both of the Hillert and Louat distributions come from mean field grain growth

theory. The mean field theory deals with the normal growth of a grain that is co

e

particles is given by the sum d the drift velocity term:

fvR

Dj +∂

f∂−= (3.7)

where f is the frequency or probability of finding a particle size R, D is the diffusivity,

and v is the velocity of particle size R. Using the continuity equation, the time derivative

f the probability function (f) can be written as:

o

)()( fvRR

fDR

jRt

f∂∂

−⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=−∂∂

=∂∂ (3.8)

In Eq. (3.8), Hillert [6] considered only the drift velocity term ( )( fvf ∂−=

∂ ) Rt ∂∂

hile Louat [7] considered only the diffusion term ( ⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

=∂∂

RfD

Rtf ).

The separate distributions are illustrated in Fig. 3.6. The largest deviation from

e ideal behavior was in specimen I (the plate-like grade) and H (the non-angular grade).

w

th

74

The sta grain size an

the normalized standard deviations are in the range of 0.55 to 0.60 in all of the grades.

ndard deviations of the distributions scale linearly with the mean d

(a)

(b) Fig. 3.5 : Plots of the carbide grain size distributions. (a) The distributions of grain sizes for each grade. For comparison, Gaussian, Louat [7], and Hillert [6] distributions are also shown. (b)

Variation of the standard deviation (σ) as a function of the grain size. The normalized standard

deviations (σ/d) for each grade are almost constant (0.55-0.60).

75

Fig. 3.6 : Carbide grain diameter distributions for each grade. Each plot is based on ~2500 WC grains. Gaussian distribution is compared with the distributions of nine grades.

The results for the binder mean free paths are summarized in Fig. 3.7 and they

were compared to stereological estimates, λ1* and λ2*. It seems clear from this

comparison that λ1*, which includes the sample contiguity of the carbide phase, is the

76

correct quantity. The log-normal distribution of binder mean free paths (not shown) is

Gaussian and it is similar to the grain size distribution.

Fig. 3.7 : Comparison of the measured binder mean free path (direct measurements) and calculated binder mean free path (stereology) using stereological equations (3.4a) and (3.4b).

The contiguity measurements are linear with the volume fraction of the binder

phase, as illustrated in Fig. 3.8. Note that the contiguity is independent of the grain size.

The average diameters of specimens A and F are more than three times larger than B and

I, yet the contiguity of A and B are essentially the same, as are those of F and I. When

one considers the uniformity of the distributions of grains sizes and binder mean free

paths, the independence of the contiguity on grain size is not too surprising. Since the

normalized size distributions in all of the microstructures are approximately the same,

topological properties such as the contiguity should also be scale invariant. While this

3.3.2 Contiguity of carbide phase (C)

77

conclusion is consistent with German’s predictions [9], there are some reports in the

literature which suggest that contiguity increases with grain size and depends on the

sintering process [10]. However, we note that the apparent contiguity would be expected

to decrease in fine-grained materials as the tail of the size distribution falls below the

detection limit of the microscopic probe.

3.3.3 Carbide crystal shape

While the contiguity is not closely linked to the average grain size, it can

potentially vary with the crystal shape. One aspect of a crystal’s shape that can be

characterized is the angularity, which we take to be inversely related to the number of

vertices bounding a polygonal particle. In other words, a wedge shaped crystal with three

vertices in a section plane has the maximum angularity. Since each carbide grain has

been approximated as a polygon, we use the average number of vertices per grain (N) as a

Fig. 3.8 : Variation of the contiguity of carbide phase as a function of the carbide volume fraction.

78

measure of the angularity. Since N has an inverse relationship with angularity, N can be

thought of as a measure of inverse angularity. As can be seen in Table 3.3, the average

umber of vertices for most of the grades varies between 6 and 7, but the inverse

angularity of grade H is almost 9. The distributions of inverse angularity are shown in Fig.

3.9(a). Separate distributions for each grade are given in Fig. 3.10. Based on this

comparison, grade H clearly has a lower angularity than the others. The measurements

exhibit a weak dependence on contiguity, as illustrated in Fig. 3.9(b). The dependence is

most easily attributed to impingement effects. Since the contiguity scales linearly with

the carbide volume fraction, the number of vertices increases as the amount of binder

decreases. As crystals impinge on one another more frequently, they have more vertices.

The average three-dimensional shapes of the carbide crystals have also been

hese measurements were conducted using a stereological technique [11, 12]

at will be described in Section 4.3. In all cases, it was found that the average particle

shape is a trigonal prism, with three

n

determined. T

th

0110 prism facets and two 0001 facets (see Fig.

2.3(a)). With one exception, the crystals are approximately equiaxed. The ratio of the

crystal’s height to the base varied from 1.2 to 1.5. The one exception was sample I, which

has an aspect ratio of 6. It should be emphasized that these average shapes are derived

only from the WC interfaces in contact with Co. In reality, the shapes of individual

crystals are always truncated in one way or another because of impingement. It is

interesting to note that the aspect ratio is linked to the contiguity and/or volume fraction

of carbide phase (except specimen I, the plate-like grade). The carbide shape is more

equiaxed in a more Co volume fraction sample. This result will be presented in Section

4.3.2 (see Fig. 4.10). The mechanical properties are apparently unaffected by the aspect

79

ratios, since the properties of specimen I (with an aspect ratio 6) do not differ

significantly from predictions based on measurements of materials with equiaxed

carbides (this will be discussed in Section 4.4).

(a)

(b) Fig. 3.9 : Number of vertices. (a) Distributions of the number of vertices per grain for each specimen grade, (b) variation of average number of vertices as a function of contiguity.

80

Fig. 3.10 : Distributions of the number of vertices for each grade. All distributions are based on ~2500 WC grains.

While no grain orientation texture was detected in these specimens, there were

significant features in the misorientation texture. All the specimens had a high population

of WC/WC grain boundaries that have a 90° twist misorientation about the

3.3.4 Orientation and misorientation distributions (see Section 4.4 for details)

]0110[ axis

81

(see Fig. 4.16(a)). Stereological analysis of these boundaries indicates that the most

common habit plane for these boundaries is 0110

ng. The specim

al boundaries in all stra

ible to co

portant param

. Because of the high planar

coincidence at this particular interface, it has been argued to be a low energy boundary

that resists wetting by liquid Co during processi ens also have an elevated

population of boundaries with a 30° misorientation about the [0001] axis (see Fig. 4.16 (b)

and (c)).

A quantitative analysis of the population of these boundaries has shown that there

are approximately constant fractions of speci ight angular grades;

90° and 30° boundaries occupy 11~14 % and 2~3 % of the total WC/WC boundaries,

respectively. Currently, therefore, it is not feas rrelate the contents of these

special boundaries with properties. However, if crack initiations at WC/WC boundaries

and/or crack propagation along these WC/WC boundaries (intergranular fracture) are

important factors for determining the failure of the composites, the populations of those

strong special interfaces remain as an im eter to potentially enhance the

mechanical properties of WC-Co composites.

.4 Correlation with mechanical properties

nation

3

The data in Table 3.2 and Table 3.3 reflect the well-known inverse correlation

between hardness and fracture toughness. Also, experimentally measured transverse

rupture strength (TRS, fracture strength) values are listed. In seeking correlations with the

microstructural parameters, we must first decide which parameters have a demonstrated

influence on the mechanical properties. An exami of the correlations indicates that

82

the angularity and aspect ratio do not influence the hardness, toughness or fracture

strength (TRS) in a significant way. On the other hand, the grain size, contiguity, binder

mean free path, and carbide volume fraction do correlate with the properties of the

materials, although the last three of these quantities are not independent. The contiguity is

linearly

related to the binder volume fraction (see Fig. 3.8), and the binder mean free path

is determined entirely by the carbide volume fraction and contiguity. Therefore, we limit

the analysis to the dependence of hardness and toughness on average carbide diameter (d)

and contiguity of carbide phase (C).

3.4.1 Hardness

For the hardness, we assume that the hardness and yield strength are related and

propose an empirical expression in analogy to the Hall-Petch relation:

321 aCad

aH ++= (3.9)

1 2 3

s a test of the generality of the analysis, we fit only the properties of grades A through G.

Grades

where the coefficients a , a , and a are determined by generalized least squares fitting.

A

H (the non-angular, coarse grained grade) and I (the plate-like, fine-grained grade)

represent excursions from the others in terms of the particle aspect ratios, grain size,

angularity, and composition. Therefore, correlations with the properties of grades H and

I can then be used as a test of the generality of the expression. Based on the grades A~G,

the fitted hardness equation as a function of WC grain diameter (d) and contiguity (C) is:

72621961057/, 2 C ) mmkgH (HV −+= , (0.9 < d < 7 [µm], 0.7 < C < 0.9) (3.10) d

83

where d has the units of [µm]. In Fig. 3.11, measured Vickers hardness (HV) of nine

grades is compared with the calculated hardness using Eq. (3.10). Also, the relative errors

are plotted at the bottom. The fitting results for grades H and I, as well as grades A-G,

show good agreement with the measured hardness; the relative errors are within 9 %.

This result suggests that the empirical equation (Eq. (3.10)) can be applied to predict

hardness in a wide range of microstructures, including those with WC particle diameter

d carbide volume fraction of 0.7-0.9, which covers most of the

comme

r) and/or alloying

lements (grade I).

d equation is asc

icrostructures. The similarities of grain size distributions of each grade and the strong

ave some physical rationale, they also include empirical constants (see Eq. (2.28)~(2.31))

[1, 4, 14, 15], and they are thus comparable to the current app

erformance of Eq (3.10) is most likely connected to the improved measurements to

which the expression was fit, not the functional form. Therefore, we expect that if these

of 0.9-7 µm, an

rcial WC/Co products. Furthermore, the equation is useful even in composites

with different WC/Co interfacial characters (grade H, non-angula

e

The success of the fitte ribed to the homogeneity of the

m

relationship between contiguity (C) and volume fraction (fc) yield to a simple linear

fitting equation consisting of two parameters, carbide grain diameter (d) and contiguity

(C).

To test the empirical hardness equation (Eq. (3.10)) and to compare with other

models, we applied the microstructural parameters of nine specimen grades to the past

models [1, 4]. The results and relative errors are shown in Fig. 3.12. Fitting the hardness

by Eq. (3.10) shows better performance than the previous models. While the past models

h

roach. The superior

p

84

empirical constants were modified to reflect more accurate measurements of the

microstructural characteristics, the predictions would improve and be similar to the

expression presented here.

Fig. 3.11 : Comparison of the calculated hardness by empirical equation with experimentally

measured data.

85

Fig. 3.12 : Comparison of the calculated hardness by empirical equation and the previous models with experimentally measured data.

3.4.2 Fracture toughness

ess, and propose following formula:

For fracture toughness, we assume a strong inverse relation between hardness and

fracture toughn

32

1 aCadaK IC ++= (3.11)

where the coefficients a1, a2, and a3 are also fitting constants. Again, to test the generality,

we found the empirical coefficients using measurements for grades A through G.

86

98.832.859.5)( −+=C

dmMPaK IC , (0.9 < d < 7 [µm], 0.7 < C < 0.9) (3.12)

The results and relative errors are given in Fig. 3.13. As is the case with hardness,

the fitted equation for fracture toughness produces results coinciding with the

experimental data. The fact that fracture toughness of grades H and I also agree with the

experimental value validates the generality of fitting equation. The relative errors are

within 9 %. Again, microstructural homogeneity (strong linearity between contiguity and

volume fraction, and the scale independent carbide grain size distribution) is assumed to

be the origin of the success of the fitted equations. The homogeneity allows us to predict

fracture toughness based on two basic microstructural parameters (WC grain diameter

and contiguity). The primary difference between the empirical equation and previous

models is that the binder mean free path is the most important parameter in the past

and contiguity. It is, of course, possible to include the

inder mean free path, but this is not an independent variable. Again, the empirical

pplicable to a wide range of microstructures with WC grain

diamete

l equations, we compared the fitting

sults with those of past models [2, 3, 5]. As can be seen in Fig. 3.14, it is clear that the

past models. Using E

easurements of grain size and contiguity make it feasible to estimate the hardness and

models [2, 3, 5, 15-17]. In the present equation, the effect of binder mean free path is

incorporated indirectly in the size

b

equation presented here is a

r of 0.9-7 µm and carbide volume fraction of 0.7-0.9.

To test the relevance of the current empirica

re

results are superior to those of q. (3.10) and Eq. (3.12), accurate

m

fracture toughness simultaneously.

87

Fig. 3.13 : Comparison of the calculated fracture toughness by empirical equation with experimentally measured data.

88

Fig. 3.14 : Comparison of the calculated fracture toughness by empirical equation and the previous models with experimentally measured data.

3.5 Summary

Atomic force microscopy (AFM) and orientation imaging microscopy (OIM)

have been used to determine the WC/WC boundary and WC/Co interface positions in

WC-Co composites. Direct measurements of the carbide and cobalt area fractions show

that the distributions of carbide grain size and binder free path are log normal and

Gaussian, and that there is a very strong linear correlation between the volume fraction of

the binder and the carbide contiguity. The number of vertices per carbide grain is slightly

89

influenced by the Co volume fraction. Empirical expressions involving the two

microstructural parameters with the most significant effect on properties, the grain size

and the carbide contiguity, fit well with the experimentally measured hardness and

fracture toughness data.

3.6 References

[1] H.C. Lee and J. Gurland, “Hardness and Deformation of Cemented Tungsten

Carbide”, Mater. Sci. Eng., 33, 125 (1978)

[2] J.R. Pickens and J. Gurland, “The Fracture Toughness of WC-Co Alloys Measured

on Single-edge Notched Beam Specimens Precracked by Electron Discharge Machining”,

Mater. Sci. Eng., 33, 135 (1978)

l and H.F. Fischmeister, “On the Fracture Toughness of Cemented Carbides”,

] H. Engqvist, S. Jacobson, and N. Axen, “A Model for Hardness of Cemented

Carbides”, Wear, 252, 384 (2002)

handran, “Fracture Toughness of Two Phase WC-Co Cermets”, Acta

etall. Mater., 42, No. 1, 143 (1994)

[6] M.

[3] L.S. Sig

Acta Metall., 36, No. 4, 887 (1988)

[4

[5] K.S. Ravic

M

Hillert, “On Theory of Normal and Abnormal Grain Growth”, Acta Metall., 13,

No. 3, 227 (1965)

[7] N.P. Louat, “Theory of Normal Grain Growth”, Acta Metall., 22, No. 6, 721 (1974)

[8] H.V. Atkinson, “Theories of normal grain growth in pure single phase systems”, Acta

metall. 36, No.3, 469 (1988)

90

[9] R.M. German, “The Contiguity of Liquid-Phase Sintered Microstructures”, Metall.

Trans. A, 16, No. 7, 1247 (1985)


Mater. Rev., 33, No. 3, 151 (1988)

1] D.M. Saylor and G.S. Rohrer, “Determining Crystal Habits from Observations of

., 85, 2799 (2002)

2] C.-S. Kim and G.S. Rohrer, “Geometric and Crystallographic Characterization of

el for hardness in WC-Co composites”, Acta

nal Plansee Seminar, Reutte, Austria, vol.2, 273 (1993)

bides”, Trans. Metall. AIME, 245, 1457 (1969)

s. A, 11A, 141 (1980)

9 (1987)

[1

Planar Sections”, J. Am. Ceram. Soc

[1

WC Surfaces and Grain Boundaries in WC-Co Composites”, Interface Sci., 12, 19 (2004)

[13] M.T. Laugier, “A microstructural mod

Metall., 33, No.11, 2093 (1985)

[14] B. Roebuck, E.G. Bennett and M.G. Gee, “Grain size measurements methods for

WC/Co hardmetals”, 13 Internatio

[15] H. Doi, Y. Fujiwara and K. Miyake, “Mechanism of plastic deformation and

dislocation damping of cemented car

[16] M. Nakamura and J. Gurland, “The fracture toughness of WC-Co two-phase alloys –

A preliminary model”, Metall. Tran

[17] L.S. Sigl and H.E. Exner, “Experimental study of the mechanics of fracture in WC-

Co alloys”, Metall. Trans. A, 18A, 129

91

Chapter 4

istributions in WC-Co composites

f observation, preferred habit planes of binder/carbide

on 3.1) are used to reveal the interface

character types of WC-Co composites. The distribution of WC/Co interfaces were

explored for all nine grades (grades A~I) and the distribution of WC/WC boundaries have

been investigated mainly for the seven straight angular grades (grades A~G).

Analysis of Interfaces in WC-Co Composites

In this chapter, a stereological method for the quantitative analysis of the

binder/carbide and carbide/carbide interface character d

is described. To quantify the types of interfaces in WC-Co composites using stereology,

geometrical and orientation data were obtained using atomic force microscopy (AFM)

and orientation imaging microscopy (OIM). Based on the stereological analysis of the

interface traces on the plane o

interfaces have been determined, and the quantitative surface area ratio of habit planes

was calculated. By the measurements of WC grain boundary misorientations, we

demonstrate that there are preferred misorientations and habit planes also in the

carbide/carbide boundaries.

4.1 Samples

The same nine specimen grades (see Secti

92

4.2 Stereological method for interface characterizations

of two-

es of

such cr

istribution of WC/WC grain boundary planes.

verview [1-7]

i) Basic

segment is

nknown, it is certain to belong to the set of planes (nijk) whose normals are

erpendicular to the segment (see Fig. 4.1(c)). If the crystal orientation (g) is known, then

The shapes of three-dimensional crystals are usually observed in the form

dimensional planar sections in micrographs. To resolve the three-dimensional shap

ystals, several techniques might be applied. Among them, serial sectioning is the

most commonly used technique. It is, however, labor-intensive, time-consuming and

resolution-limited when the sectioning is accomplished by polishing. To make three-

dimensional images of crystals, 3DXDM (three-dimensional X-ray diffraction

microscopy) can also be used. However, this apparatus is still in development, not yet

widely available, and also has limited resolution. In this thesis, we used a recently

developed stereological method [1-7] to determine the three-dimensional shapes of WC

crystals embedded in a Co binder. This method was applied not only to estimate the WC

habit planes (for investigating the three-dimensional form of carbides) in contact with Co,

but also to characterize the d

4.2.1 Method o

idea

In two-dimensional micrographs, sections of faceted crystals (WC crystals)

appear as polygons, where each line segment, lij, comes from a different bounding plane

(see Fig. 4.1(a)). While the actual plane that created the observed line

u

p

93

the possible habit planes can be transformed to the crystal reference frame, n'ijk=gnijk. On

stereographic projection, the planes n'ijk lie on a great circle perpendicular to the

position

a

of the line segment (see Fig. 4.1(d)). By observing many segments from different

crystals, a corresponding number of sets of possible planes (each on a different great

circle) that might be responsible for the observations are also identified. These

observations are used to calculate the probability, p(n'), that a given length of line on the

perimeter of a random section plane falls on a plane with the orientation n' according to

the following equation:

∑

∑= kji ,,

||)(

ijln'p (4.1)

The key to this analysis is that the correct habit planes are sampled with a high

probability: there is at least one habit plane in each set of possible planes, n

kji

i

,,

|| ijijk lng

e positions of

ue habit planes and p(n') will exhibit local maxima at these positions. Because non-

rous, any specific non-habit plane is observed much less

equently than a correct habit plane.

ijk. Therefore,

the great circles arising from each observed line segment must intersect at th

tr

habit planes are far more nume

fr

94

(a) (b)

total length of a set of randomly distributed lines

tersecting an area is proportional to that area, the ratios of the line lengths associated

with each habit plane provide an estimate of the relative surface area [8]. To determine

which line segments are associated with each habit plane, we compute the angle (α)

between each line segment, l'ij, and each previously identified habit plane, n'hkl, in the

crystal reference frame:

α = cos-1[l'ij • n'hkl/(|l'ij| |n'ijk|)] (4.2)

Fig. 4.1 : Schematic illustration of stereological analysis (see text for details).

ii) Trace classification

To determine the crystal shape, it is necessary to know the relative area of each

habit plane. Since it is known that the

in

(c) (d)

lij

nijklij

nijk

i = 1xl

z

yl12 l13

11

φ

θk

x

z

y

f

l

nijk

ij

95

tion

and experim

segm

e of a

small set of es. After all of the data have been classified in this way,

which we call the trace classification procedure, we have the total length of line per habit

plane and ratios of these quantities yield the su e area ratio.

Here, we apply this method to both WC/WC grain boundaries and WC/Co

interfaces for the two-parameter calculations. For the WC/Co interfaces, the boundary

trace, nijk, is transformed to the WC reference frame. For the WC/WC boundaries, the

planes in the zone of the boundary trace, nijk, are transformed to the crystal reference

iii) Applicability and limitations of stereological analysis

There are two essential preconditions in order to apply the stereology described

above. First, polycrystals must be randomly oriented with respect to the sample reference

frame. If there is strong orientation texture, the analysis will be biased by the texture.

Second, the boundary planes should have random inclinations with respect to the sample

surface (see Fig. 4.2 (a)). Since we assume a random distribution of inclinations in the

sample reference frame, any deviation from this will create error in the result. Sometimes,

most of grain boundaries are parallel to sample normal direction (see Fig. 4.2 (b)). This is

a so called ‘columnar boundary structure’. In this case, it is not necessary to use the

For the habit plane of origin, |α-90°| ideally vanishes. In practice, discretiza

ental errors lead to finite values of |α-90°|. Therefore, we assign a line

ent to a habit plane whenever |α-90°| is less than some predetermined tolerance

angle, between 3 and 10°. This condition allows us to assign each line length to on

possible habit plan

rfac

frame once with each of the orientations of the two adjoining crystals.

stereological method to determine grain boundary distributions; knowledge of the grain

96

orientations, the boundary trace, and the boundary plane normal vector lies in the surface

plane completely specifies all the five parameters that describe a grain boundary.

inclinations, (b) columnar boundary structure.

To use the stereological method, an extensive data set of line segments must be

ximate resolution of 10° (see

(a)

(b) Fig. 4.2 : Schematic cross-sections of boundary structures. (a) Boundary structure with random

acquired. The analysis is, therefore, limited in an appro

Section 4.2.2 ii) for details, sometimes, we can use a small resolution angle in the

misorientation averaged calculations). However, since there exist an enormous number of

different types of grain boundaries and the number increases as resolution angle

decreases (see Fig. 4.3, there are ~104 to 105 distinct types of boundaries with 10°

resolution depending on crystal symmetries), 10° resolution remains within a reasonable

angular mesoscale range.

97

102

103

10

10

10

108

10

10

191715131197531

9

10 cubic

5

6

7

104

tetragonal hexagonal

type

s of

bou

ndar

ies

resolution angle (o)

.2.2 N

segment), and by ‘automatic extraction’ in OIM images using the OIM software

(TexSEM Laboratories, Inc. version 3.03) following the method outlined by Wright and

Larsen [9]. The line segment data using the manual method is more accurate in

positioning exact locations of interfaces but it has limitations in time-and-labor. On the

other hand, the line segment data by the automatic method are easily obtained, but less

Fig. 4.3 : Different types of boundaries as a function of resolution angles (in degrees). Number of boundary types increases as resolution angle decreases.

4 umerical analysis

i) Two- and five-parameter calculations

All line segments for the analysis of interfacial plane distributions were obtained

from two methods; by ‘manual hand tracing’ of boundary positions in AFM images using

a computer mouse (a program automatically records the vector components of each line

98

accurate. Regardless of how they are obtained, each segment is then as

Euler angles obtained from the OIM image. In this work, we first use

anual tracing based on AFM images, and then applied the line seg

atic extractions through OIM images.

Among the five macroscopic parameters of grain boundaries, the stereological

thod basically approximates one parameter that describes grain boundary plane

ations. The other four parameters (three from misorienta

boundary planes) can be easily identified in two-dimensional mi

this stereology, therefore, we can either ignore (two-parameter calculation, m

sociated with a set

of d data set from

the m ment data from

the autom

me

inclin tions and one from

crographs. In applying

isorientation

ed calculation) or include (five-parameter calculation) the three parameters

distributions in the two- and five-

arameter calculations are represented by the notations λ(n) and λ(∆g, n), respectively,

urrences of a grain boundary. Here, ∆g and n denote a

isorientation and grain boundary plane normal, respectively (see Section 2.2.1 ii)). For

ulations, the spherical domain of planes in both

the lab

averag

explaining misorientations. The grain boundary

p

which are the relative occ

m

both of the two- and five-parameter calc

oratory and crystal reference frames was partitioned into equal area cells with the

in-plane angle, φ, in the range between 0° and 360°, and the out-of-plane parameter, cosθ,

in the range of 0 to 1 (see Fig. 4.4). In the case of five-parameter calculation,

misorientation space must also be discretized. Here, the three Euler angles were used to

parameterized the misorientations in the range ∆φ1=0° to 360°, cos(∆Φ)=0 to 1, and

∆φ2=0° to 360° so that the volume of each cell in the Euler angle space is equal. Hence,

the numbers of cells are 2

90360∆× (two-parameter calculation) and

99

5

90360360180360∆

×××× (five-parameter calculation), where ∆ is the resolution angle

(in degrees). The discretized domain forms ‘accumulator matrix’, in which the observed

boundary trace, nijk, is added. For the calculation of misorientation averaged boundary

plane distributions (two-param

6

the total domain is shown. The out-of-plane angle cosθ ranges from 0 to 1, and the in-plane angle

eter calculation), there were two dimensional arrays of 324

(=36×9), 1296 (=72×18), or 3600 (=120×30) equal cells, depending on approximate

resolution of 10°, 5° or 3°. The five-parameter analysis was conducted using a resolution

of 10° which results in five dimensional arrays of ~7.5×10 (=36×18×36×36×9) equal

volume cells. The symmetry operators for the hexagonal system were applied to all of the

observed line segments in the crystal reference frame.

Fig. 4.4 : Illustration of discretized domain for the analysis of binder/carbide interface. Only ½ of

φ ranges from 0 to 2π.

φ

θ

100

ii) Number of line segmen

For a reliable and smooth plane distribution, experience has shown that each

ts

distinguishable cell should contain on average of 100 (two-parameter calculation) and 10

(five-parameter calculation) line segments. Thus, the number of line segments that are

required in the analysis is given by:

N2 = 100903602 ×

×× O∆

(two-parameter calculation) (4.3a)

N5 = 102 25 ×

××36018036090360 ××××

O∆ (five-parameter calculation) (4.3b)

where N2 and N5 are the respective minimum number of line segments for two- and five-

parameter calculations, and O is the number of proper symmetry operators for a given

crystal system. For example, in a 3 line segments are

required for ∆=10, Ο=12), and

~2.6×105 segment with a resolution of

10° (∆=10, Ο 3 (two-parameter,

∆=10, Ο=24), 6.5×10 , and they increase

to 8.2×103 ter, ∆=10, Ο=4) in a

rom this, it is obvious that the five-parameter calculation using the

carried out based on both of the manual and automatic schemes, but the five-parameter

lculation resorted to only the automatic method.

It should be also noted that the above approximations (Eq. (4.3a) and (4.3b)) are

r the case of analyses from the automatic tracings. When the manual tracings are used

hexagonal system, at least ~2.8×10

the two-parameter calculation with a resolution of 10° (

s are required for the five-parameter calculation

=12). Those number requirements decrease to 1.4×10

5 (five-parameter, ∆=10, Ο=24) in a cubic system

(two-parameter, ∆=10, Ο=4), 2.4×106 (five-parame

trigonal system. F

manual tracings is not feasible. Thus, in this work, the two-parameter analysis was

ca

fo

101

in the two-parameter calculations, it turned out that the result exhibits a faster

of analyzed line segments increases, the aspect ratio

onverges asymptotically and reaches a constant value. The variation of aspect ratio in

segments need to be analyzed. Here, all the results from manual tracings are based on at

least 1000 line segments.

convergence to a smooth distribution. For example, we calculated the aspect (base-to-

height) ratio of grade G as a function of the number of line segments included in the

analysis (see Section 4.2 for the detailed description of aspect ratio calculation). In Fig.

4.5, it seems clear that as the number

c

Fig. 4.5 suggests that, for reliable results, more than ~1000 WC/Co interface line

Fig. 4.5 : Variation of aspect ratio with the number of line segments included in the calculations (for grade G).

102

iii) Background subtraction

Since every observed boundary trace generates a ser of possible plane orientations

in the crystal reference frame, it is apparent that too much length has been added to the

cells of the accumulator matrix. If the number of possible discrete boundary plane

orientations in the crystal reference frame is D (= 2

90360∆× , where ∆ is a resolution angle),

the observed length ( il ) at i boundary plane cell is composed of the correct length ( il )

from this cell (i) and the incorrect length ( ICil ) associate with some other cell (j, j=1~D,

j≠i). In other words, ICil is the quantity that has been added to the

O th

ith cell but actually

e from jth cell (j, j=1~D, j≠i). Now, we define a weighting factor (ωij) that implies the

fraction of traces whose correct boundary plane is j, but also pass through i because they

are in the same zone:

(4.4)

Since the weighting factor, ωij, is largest when i and j are neighboring cells and

decreases to a constant value as the angular separation between i and j increases, we

divide the incorrectly assigned length, , into two parts. The first part originates from

eighbor cells and the second part comes from non-nearest neighbor cells. If we

eighbor as the average correct length

C

cam

∑≠=

+=+=D

ijj

Cjij

Ci

ICi

Ci

Oi lllll

1ϖ

ICil

nearest n

approximate the incorrectly assigned length in the jth cell originating from non-nearest

n ( Cl ) in each cell, then the incorrectly assigned

length in the ith cell is then given by:

103

∑∑ ≠=

≠=

ijj

ijj 11

the number of cells on a great circle. Then, the average of incorrectly assigned length

from non-nearest cells can be written as:

≈D

Cij

DCjij ll ϖϖ (j≠nearest neighbors of i) (4.5)

For any accumulated length, we assume that the probability of a correct

assignment is 1/B and the probability of an incorrect assignment is (B-1)/B, where B is

OD

ijj

C

BB(

1

−

≠=

ij ll )1=∑ϖ (j≠ nearest neighbors of i) (4.6)

Now, taking into consideration that the ratio of Ol to Cl is B:

(4.7)

For the incorrectly assigned length from nearest neighbor cells, we assume that

e leng

approximation is valid as long as the number of cells on a great circle (B) is large enough.

Combining Eq. (4.5) ~ Eq. (4.7) yields:

11

−=∑≠=

BD

ijj

ijϖ

th th in a cell is approximately equal to the length in a nearest neighbor cell. This

CONN

ijj

CD

NNj

CD

ijj

CIC lBZlBZllll )1()1()1(111

−+−

−≈+== ∑∑∑≠=+=

≠=

ϖϖϖ (4.9)

Finally, we reach the following expression:

iijijjiji B

104

)1(1

)1()1(

−+

−−+

≈BZ

l O

BBZl

lOi

Ci (4.10)

) Data processing; OIM image clean-up

grain boundary misori

images. The orientation data were processed to remove spurious observations using a

exSEM Laboratories, Inc. version 3.03).

efore the clean-up procedure, the crystallographic orientation of the Co phase has been

, and φ2=0,

diffraction patterns of the Co phase. There are two advantages to this false assignment.

irst, it

single orientation was assigned to each carbide

rain, by averaging all of the orientations within a single grain that were measured with a

ence index (CI) greater than 0.2. An example of automatic line segment

is shown in Fig. 4.6. In the cleaned-up IPF maps (Fig. 4.6 (b) and (c)), the Co phase is

olored 0, Φ=0, and φ2=0 (basal orientation).

or the comparison of boundaries between original and cleaned-up maps, an image

quality (IQ) map is also illustrated. The orientations of individual adjoining grains (ga and

Bear in mind that the final equation (Eq. (4.10)) can be applied in a smooth and

continuous distribution. The detailed description for this ‘background subtraction’ is

available in reference [5].

iv

The distribution of entations was determined from OIM

‘grain dilation clean-up’ in the OIM software (T

B

fictitiously assigned as φ1=0, Φ=0 since it was not possible to get reliable

F makes it easy to distinguish the carbide and binder phase. Second, because the Co

phase is indexed as a single orientation, we avoid distorting to WC/Co interface positions

in the clean-up procedure. This second advantage is especially important when the line

segments are extracted automatically. A

g

confid extraction

as red because it has the orientation of φ1=c

F

105

gb) were then used to calculate the misorientation (∆gab = gagbT) across each boundary.

tion angle (ω = cos-1 ii

tation axis (ri = -εijk∆gjk/(2sinω)) were calculated. Note that the calculation of

misorientations is useful only for the WC/WC boundaries. In the distributions presented

here, the relative population of boundary types is weighted by boundary length.

In the next two sections, the results for WC/Co interfaces and WC/WC

boundaries will be discussed separately. As examples, grades E and A will be discussed

in detail and compared to summary results from the other grades for the WC/Co and

WC/WC interface distributions, respectively.

From the misorientation, the rota [(∆g -1)/2]) and components of

ro

106

Fig. 4.6 : Example of automatic extraction of line segments (from grade A). (a) original IPF map, (b) cleaned-up IPF map, (c) cleaned-up IPF map with extracted line segments, (d) IQ (image quality) map.

(a) (b)

(c) (d)

107

4.3 WC/Co interfaces

The habit planes of the WC/Co interfaces are formed by the interaction between

the liquid phase (Co) and the solid phase (WC) during the coarsening process. Here, the

results from the manual tracings are described in detail, followed by a discussion of the

results from the automatic extractions. Note that only the two-parameter calculation is

relevant for the analysis of WC/Co interfaces.

4.3.1 Analysis of grade E

i) Result from the manual tracings

In this section, the WC/Co interface distribution, λ(n), of grade E (grain diameter

1.40 µm, and carbide volume fraction 0.6919, the low carbide volume fraction grade) that

results from the analysis of line segments obtained by manual tracing will be described.

The stereographic projections in Fig. 4.7 show plots of WC/Co interface distribution,

λ(n), determined from (a) 50 observed crystals (~160 line segments) and (b) 200

observed crystals (~670 line segments) of grade E with an approximate resolution of 10°.

As mentioned earlier, this is the probability that a length of line on the perimeter of an

observed section of a WC crystal plane was created by a plane with a specific orientation:

the results are presented in multiples of a random distribution (MRD). In both cases, the

peaks at the 0001 and 101 0 positions demonstrate that the basal and prism planes

dominate the WC crystal habit. To determine what fraction of the observations belong to

ach habit plane, we computed the angle (α) between each line segment (in the crystal e

108

reference frame) and the normals to 0001 and 101 0. For |α-90°| ≤ 5° (5° tolerance),

0% of all the observed line segments belong to one of the two habit planes. If the

toleran

8

ce angle is increased to 10°, then the fraction of observations that can be assigned

to either the 0001 and 101 0 planes increases to 92%. When the 5° tolerance angle is

used, which is comparable to the anticipated experimental errors, the basal-to-prism facet

area ratio is 0.34. This ratio is not strongly sensitive to α; it varies from 0.31 to 0.35 as

the tolerance angle is varied from 3° to 10°.

Fig. 4.7 : WC/Co Habit plane probability distribution derived from (a) 50 and (b) 200 WC crystals of grade E. Both are plotted in [0001] stereographic projection. represents a basal

Co is a hexagonal prism with six indistinguishable 10

(a) λ(n) (b) λ(n)

position and stands for a prismatic position.

The results suggest that the average three-dimensional shape of the WC crystal in

1 0 prism facets and two 0001

basal facets. To test this assignment, we can inspect the shape of the sections that happen

109

to be oriented parallel or perpendicular to the basal plane. Sections perpendicular to the

basal plane should be rectangular and those sections parallel the basal plane should be

hexagonal. As illustrated in Fig. 4.8, those sections perpendicular to 0001 are

rectangular, as expected. However, those parallel to the 0001 plane are invariably

triangular; hexagonal sections of WC crystals parallel to 0001 are not observed. This

indicates that the shape is a trigonal prism bound by only three of the six

indistinguishable 101 0 planes. Considering this shape and the area ratio cited above,

e trigonal prism is roughly equiaxed, with an edge length that is 1.18 times the height,

as illustrated in Fig. 4.9. This shape is consistent with earlier observations reported for

similar alloys [10-13]. However, it has to be noticed that the shape in Fig. 4.9 is not an

equilibrium form. It is an illustration of the relative surface area of each facet and it

should be regarded as an average shape or the growth form rather than equilibrium form.

The vast majority of the WC/Co boundary traces are consistent with crystal

terminations along

th

101 0

C

this conclusion is consistent with earlier obs

is is assum

e

always in contact with other crystals and

C

and 0001 planes and from this, we conclude that the

average shape of a W crystal in Co is an approximately equiaxed trigonal prism. Since

ervations of deep etched samples [10-13], the

stereological analys ed to be reliable. It should be kept in mind that the result

from our analysis is m rely an average shape; in the composite the WC crystals are

the shape is, therefore, truncated from the

idealized form. What we can be certain of from the current results is that approximately

two thirds of the W in contact with Co has an 101 0 orientation and the remainder has

the 0001 orientation.

110

(1)

(1)

(1) (3)

(2)

(a)

Fig. 4.8 : Comparison of WC crystal shape in (a) AFM topography image and (b) OIM inverse pole figure (IPF) map showing the crystal orientation. Both sections are from grade E. Triangular crystals oriented with [0001] perpendicular to the paper are labeled (1).

Fig. 4.9 : The proposed average shape of WC crystal in Co (data from grade E).

(1)

(1)

(1) (3)

(2)

(1)

(2)

(3)

(b) 4 µm

1 prism plane

1.18

basal plane

111

The observed trigonal symmetry of the crystal habit presumably derives from the

two different terminations of the prism faces [11]. If one terminates the WC structure by

breaking the fewest number of bonds, then the atoms on the six 101 0 faces each retain

four nearest neighbors (see Fig. 4.10(a)). However, there are now two distinct sets of

prism faces; three terminated by W and three terminated by C. Either set of three can

form the observed triangular prism. As long as one of the two terminations is more stable

than the other in contact with Co, then the observed trigonal shape is expected. Based on

the current data, it is not possible to say which of the terminations is observed (see Fig.

4.10(b) and 4.10(c)).

ii) Result from the automatic extractions

ents (

In Fig. 4.11 (c), the interface plane distribution, λ(n), from the automatically

extracted line segm ~21000 segments) of specimen grade E is shown on a

stereographic projection. The results are similar to those from manual tracings, but the

peak heights differ; Peak positions are same (basal and prismatic positions), and the

relative MRD ratios of basal to prismatic facets are approximately same. Because the

manual and automatic results are comparable, the automatic method is assumed to have

sufficient accuracy.

112

(a)

W atom

terminated by C

C atom

(b) Fig. 4.10 : Atomic models C prism planes. The smaller shaded circles represent C and the larger open circles represent W. (a) Hexagonal shape with both C and W terminated surface, and (b), (c) two possible trigonal shapes with W and C terminated surfaces, respectively.

4.3.2 Analysis of other grades

Analyses of the other sp istributions dominated

different aspect

ase-to-height) ratios. Fig. 4.11 includes WC/Co interface plane distributions from

selected specimen grades (automatic extractions, data from grades B, D, E, H, and I) with

terminated by W

(c) for the chemical termination of W

ecimen grades showed interface d

by basal and prism facets (see Fig. 4.11 (a)~(e)). However, they exhibit

(b

113

an app

ic/basa

volume increases. In other words, the WC crystals in

omposites with a high Co fraction display a more equiaxed shape. It is a general trend in

s and automatic extractions.

ter than the prismatic plane.

roximate resolution of 10°. The upper three are based on straight angular grades

with different Co fractions (similar average carbide grain size). While maximum peak

heights at prismatic positions are almost the same, the prismat l peak ratios are

different. It turned out that the aspect ratio is related to the carbide volume fraction. It is

more obvious when we calculate and compare the aspect ratio using trace classification

method. Fig. 4.12 shows the estimated aspect ratio as a function of the carbide volume

fraction. Aspect ratios were calculated using both from the manual tracing and automatic

extraction methods. If we ignore the result of grade I (the plate-like grade), the aspect

ratio increases as the carbide

c

calculations both from the manual tracing

The lower two results in Fig. 4.11 are from atypical specimen grades; the straight

non-angular grade (grade H) and the alloyed grade (grade I). In non-angular grade (Fig.

4.11 (d)), maximum peak height at the prismatic facet is smaller than in the angular

grades (Fig. 4.11 (a)~(c)). Also, there was a measurable distribution of facets that were

neither prismatic nor basal. In the alloyed grade (Fig. 4.11 (e)), the peak intensity at the

basal facet is stronger than that of prismatic facet, which indicates that the relative area of

the basal plane is grea

114

(b)

grade D, (c), grade E, (d) grade H (the non-angluar grade), and (e) grade I (the plate-like grade).

(a)

(c)

(d) (e) Fig. 4.11 : WC/Co interface plane distributions, λ(n), from automatic extractions. (a) grade B, (b)

represents basal position and stands prismatic positions.

115

Fig. 4.12 : Variation of aspect ratio with the carbide volume fraction.

4.4 WC/WC boundaries

4.4.1 Analysis of grade A (two-parameter calculation)

In this section, results from misorientation averaged WC/WC grain boundary

distributions, λ(n), will be discussed for specimen grade A (grain diameter 5.31 µm, and

carbide volume fraction 0.8852, the high carbide volume fraction grade). The results

Recall that for a WC/WC boundary plane distribution analysis, we can either

ignore the misorientations (λ(n), the two-parameter calculation) or include the

misorientations (λ(∆g, n), the five-parameter calculation). This work is focused on the

two-parameter calculation, and the five-parameter calculation results are briefly discussed

in the last section (Section 4.4.3).

116 116

presented here are based on observations

boundaries in a planar section of specim

Based on an exam

WC crystals exhibit no orientation texture

sorientation texture. The misorientatio

(~15,600) has been determined and Fig. 4.

of ~8400 WC grains and ~15600 WC/WC

en grade A.

ination of (0001) pole figure of grade A, we conclude that the

(see Fig. 4.13). There is, however, a strong

mi n across all of observed WC/WC boundaries

14 illustrates the relative frequency of

rientation angles. For comparison, the

also shown [14].

Note that the experimental distribution shows two peaks, at ~30° and 90°, which are not

consistent with the random distribution. To examine the distribution of misorientation

axes, the data are plotted in axis-angle space, as illustrated in Fig. 4.15. Note that at 30°,

there is a peak at the position of the [0001] rotation axis and at 90°, there is another peak

at the position of the

occurrence of boundaries with different miso

distribution expected for randomly misoriented hexagonal crystals is

[101 0] axis, indicating that boundaries with these two

uch higher than random frequency.

Fig. 4.13 : Pole figure of WC crystals (based on ~8400 grains of grade A). No orientation textures found.

misorientations occur with a m

(0001) 0)11(2

117

Fig. 4.14 : Grain boundary population of WC crystals (grade A) and random objects as a function

We can also analyze the habit planes of the boundaries that have a 30° rotation

ose that have a 90° rotation about the

of misorientation angle.

[101 0] around the [0001] axis and th axis. Here, we

onsider the bounding plane of each crystal independently. In other words, in the analysis c

of the line segments, we are applying the complete hexagonal symmetry of each crystal.

The habit plane distribution for the crystals comprising boundaries with 90° rotations

about the >< 0110 axis, λ(∆g, n)=λ(90°/ >< 0110 , n), are depicted in Fig. 4.16(a).

Here the misorientation, ∆g, is specified by an angle/axis notation, 90°/ ]0110[ . It is clear

that the habit of this boundary is on the 101 0 plane, which is perpendicular to the

ommon rotation axis, [101 0]c . As long as the interface plane is perpendicular to the

the same in both crystals and

it is a p

bicrystal's common rotation axis, then the bounding plane is

ure twist configuration. If there was a tilt component, the boundary could not be

118

constructed from only 101 0 planes. The bicrystal formed by a 90° [101 0] twist is

illustrated schematically in Fig. 4.17(a).

Fig. 4.15 : WC/WC grain bo Sections of fundame

λ(∆g,

n)=λ(30°/<0001>, n ted in

k at the basal position confirms the presence of twist boundaries. As

undary misorientation distribution (grade A) in axis-angle space.ntal zone are taken perpendicular to the [0001] axis.

The boundaries with 30° rotations around the <0001> axis,

), on the other hand, exhibit multiple habit planes, as illustra

Fig. 4.16(b). The pea

119

mentioned above, as long as the interface plane is perpendicular to the common axis, it is

a twist boundary. This configuration is illustrated in Fig. 4.17(b). The remaining habit

planes are 101 0 and 112 0

axis and

ntary planes that com

. The normals to these planes are perpendicular to the

[0001] misorientation separated by 30° (the misorientation angle). Therefore,

they are compleme prise the same 30° tilt boundary. In other words,

when one crystal is terminated by a 101 0

ystal rotated by 30°

plane, the geometrically required

complement (parallel plane) in the cr around [0001] is 112 0 .

Therefore, both of the two habit planes perpendicular to the [0001] axis can be accounted

for by the same asymmetric tilt bicrystal, illustrated in Fig. 4.17(c).

, n)=λ(

(a) λ(∆g 90°/ >< 010 , n) λ(∆g, n)=λ(30°/<0001>, n)

grade A. Both are plotted in the [0001] stereographic projection. , , and sent (0001),

1 (b) Fig. 4.16 : Habit plane probability distribution for (a) 90° boundaries and (b) 30° boundaries of

repre

)0110( , and )0211( poles, respectively.

120

(a)

(b) (c) Fig. 4.17 : Schematic representations of the three most common types of grain boundaries in

composites. (a) 90° ]0110[ boundary with two 0110

with

planes, (b) 30° [0001] boundary with

two 0001 planes, and (c) 30° [0001] boundary 0110 and 0211 planes.

Using a tolerance of 3° on the axis and angle parameters and a tolerance of 5° on

the boundary plane parameters, we find that 11% of all the observed boundaries have the

[101 0] 90° twist configuration and 3% have one of the two 30° [0001] configurations.

oundaries. They might be left over from the starting materials or they might form when

two grains impinge as the result of either coarsening or grain growth. Neither process

One can imagine several possible mechanisms for the formation of the WC/WC grain

b

suggests a mechanism for creating higher populations of certain boundary types. A more

likely explanation is that a random distribution of boundaries is created by growth related

121

impingement and that the highest energy boundaries are then eliminated from the

population by the infiltration of Co. This process, which is consistent with the observed

random orientation texture of the samples, would lead to an enhanced population of

boundaries that are resistant to wetting [15]. The boundaries that are preserved in the

population must have energies that are less than twice the WC/Co interface energy.

Based on the special geometric characteristics of the three boundaries that

dominate the population, it seems feasible that they represent relatively low energy

configurations and resist wetting. The most populous boundary is the 90° [101 t

boundary. The WC unit cell has a c/a ratio of 0.976 so that a 90° rotation about [10

0] twis

1 0]

that brings [0001] axis in one crystal parallel to ]0112[ in the other and the misfit

etween the two rectangular surface units is less than 3%. The stability of this boundary

against wetting has been demonstrated in a TEM study, which showed a direct WC to

WC transition at this boundary [16]. The 30° [0001] twist configuration is close to a

relatively high coincidence boundary (Σ13, 28° [0001], in which one out of 13 lattice

sites are in coincidence). If we assume that the high coincidence also leads to a low

energy, then this can potentially explain its stability against wetting. However, the 30°

[0001] asymmetric tilt, while it forms the same coincident site lattice, does not have an

interface plane of high coincidence. Farooq and Klement used orientation imaging

microscopy to measure the distribution of WC/WC grain boundary. In contrast to our

results, they reported that the distribution is nearly random and that there are no

significant amounts of special boundaries [20]. However, it is considered that the

WC/WC grain boundaries that were used in their analysis are distorted by the OIM clean-

up procedure.

b

122

A second geometric principal that can be

grain boundaries is that interfaces ma

generally have lower energies [17]. The basic

spacing are also relatively flatter, and two sm

rough surfaces. The superior fit results in greate

and less repulsion, which means that the bound

case of WC, the

used to assess the relative energies of

de up of planes with large interplanar spacings

idea is that planes with large interplanar

ooth surfaces fit together better than two

r attractive forces across the boundary

ary has a relatively lower energy. For the

101 0 and 0001 interface planes have the two greatest interplanar

spacings, 2.52 Å and 2.84 Å, respectively. The 112 0 plane, which is a component of

the [0001] asymmetric tilt boundary, has the fourth largest interplanar spacing, 1.45 Å.

Therefore, if we assess the energies of the boundaries according to the interplanar

spacings of their component planes, we also conclude that the three most common

boundaries have relatively low energies.

Another potential explanation for the occurrence of a high population of

boundaries with 101 0

r

expected to align with their habit planes pa

and 0001 interface planes is that the crystals are already

terminated by these o ientations in the liquid Co and, if forced together, would be

rallel. This process would clearly favor the

occurrence of [101 0] and [0001] twist boundaries. However, since these twist boundaries

are observed preferentially at characteristic angles (90° and 30°), the crystals would also

have to rotate into these particular configurations. Considering the constraints imposed by

e other crystals in the composite, this does not seem very likely. Therefore, an

enhanced stability against wetting seems to be a more likely explanation for the high

population of these boundaries.

th

123

4.4.2 Analysis of other grades (two-parameter calculations)

The WC/WC boundary segments data for other specimen grades were extracted

from the OIM images from the TSL software. The stereological method has been applied

with an approximate resolution of 10°. 2×104~4×104 WC/WC boundary line segments,

lations. More line segments

minations of

WC/WC grain boundary plane distributions (misorientation averaged distribution, not

shown) again revealed two prominent peaks at prismatic and basal positions for all grades.

Furthermore, the peak heights and the ratios of two dominant facets in all angular straight

grades are almost indistinguishable. All of the calculated aspect ratio (base-to-height ratio)

(90°

depending on the specimen grade, were used in the calcu

were obtained from the grades with higher carbide volume fractions. Exa

falls in the range of 1.12 to 1.19 (see Fig. 4.12).

[101 0] To quantify the populations of two WC/WC grain boundaries twist

and 5° tolerance on the boundary plane parameters) as for the straight angular grades

rades A~G). The population of these two boundaries in all straight angular grades

appears to be nearly constant, occupying 11 to 14 % and 2 to 3 % for 90° and 30°

configurations, respectively; recall that the respective populations for the specimen grade

A are 11 and 3 %. The populations, on the other hand, for specimen grade H (straight

non-angular grade) and I (alloyed plate-like grade) were found to be different from the

straight angular grades. As expected, the non-angular grade (grade H) displays low

populations for the two special boundaries; 7 % for 90° boundaries and 1% for 30°

boundaries. In case of the plate-like grade (grade I), 30° boundaries exceed 90°

boundaries in population, since the basal facet is more frequent than the prismatic one;

and 30° [0001] ), we used the same tolerance angles (3° on the axis and angle parameters

(g

124

90° and 30° boundaries make up 6 % and 7 % of the populations, respectively. The

occurrences of basal and prismatic facets in WC/WC boundaries are less frequent than

for WC/Co interfaces. With a 5° tolerance angle, the percentage of the two special facets

(basal and prismatic) decreases from ~80 % of the WC/Co interfaces to ~60 % of the

WC/WC grain boundaries. The peak height at the two positions reflects this fact. Fig.

of W

). The

l interfa

(a) λ(n) (b) λ(n)

grain boundaries. represents a basal position and stands for a prismatic position.

4.17 (a) and (b) compares the plane distributions between WC/Co interfaces (Fig. 4.18 (a),

same as Fig. 4.11 (a)) and WC/WC boundaries (Fig. 4.18 (b)) in the specimen grade B.

The respective peak heights of the distribution at prismatic orientations C/Co

interfaces (MRD=4.93) are higher than for WC/WC boundaries (MRD=4.02

reduced area of special orientations for the interna ces is probably the result of

impingement that occurs during coarsening. The same phenomena was found in all

specimen grades (grades A~H).

Fig. 4.18 : Habit plane distributions of grad B from automatic extractions. (a) WC/Co interfaces,and (b) WC/WC

125

4.4.3 Five-parameter calculation

Up to now, misorientation averaged results (two-parameter calculation) for

WC/Co interfaces and WC/WC boundaries were discussed. To include misorientations in

the WC/WC grain boundary plane distribution analysis, it is necessary to have at least

2.6×104 line segments (see estimates in Section 4.2.1). Since the populations of the two

special boundaries in each straight angular specimen grade are almost invariant (11 to 14

% and 2 to3 % for 90° 0110 and 30° 0001 boundaries, respectively), we have

combined all of the data from all of the straight angular grades. The total number of line

segments accrued was ~2.2×104, which is approximately 85 % of the required number.

While some of the misorientations will not have enough data for a reliable analysis, data

from the highly populated misorientations should be sufficient fro the five-parameter

analysis.

Fig. 4.19 (a)~(c) illustrate the grain boundary plane distributions, λ(∆g, n), for

misorientations of 90° about the ]0110[ axis, 30° about the [0001] axis, and 90° about

the ]1102[ axis. A strong peak at the )0110( position in Fig. 4.19 (a) confirms the 90°

twist configuration and peaks at the 0110 , 0211 and (0001) positions in Fig. 4.19 (b)

demonstrate the 30° twist and 30° asymmetric configurations. It appears that the 30° twist

boundary is more populous than the 30° asymmetric boundaries in general straight

angular grades; the MRD value at the (0001) orientation is 18.2 and the MRD value at the

0110 and 0211 orientations is ately four. Note that the grain boundary

istribution (λ(∆g, n)=λ(90°

approxim

]0110[ , n), five-parameter calculation) in Fig. 4. 19(a) is d

126

different from the distribution (λ(∆g, n)=λ(90° >< 0110 , n), two-parameter calculation,

l of tal he 90° boundaries were used) in Fig. 4.16 (a). After spanning all misorientation

space, we found another set of boundaries other than these two special misorientation-

boundary pairs that have relatively high populations at 90° about ]1102[ (see Fig. 4.19

(c)). The peaks at the basal and prismatic orientations (which are separated by 90°)

indicate the presence of an asymmetric tilt boundary.

The fact that there is a strong preference for low index planes in WC/WC grain

boundaries is consistent with observations in other polycrystalline materials [3, 4, 6].

Even if there is a thin layer of Co between the carbide grains as some has argued [18, 19]

(dispersed carbide hypothesis, see Section 2.2.2), it apparently does not greatly affect the

tendency for specific low index interface planes to align with one another. The carbide

skeletons are connected mostly by the 90° and 30° misorientations and these boundaries

are thought to be low energy configurations.

127

(a) λ(∆g, n)=λ(90°/ ]0110[ , n) (b) λ(∆g, n)=λ(90°/[0001], n)

(c) λ(∆g, n)=λ(90°/ ]1102[ , n)

Fig. 4.19 : Grain boundary plane distributions. (a) 90° misorientation about ]0110[ axis, (b) 30°

misorientation about [0001] axis, and (c) 90° misorientation about ]1102[ axis. , , and

represent (0001), )0110( , and )0211( poles, respectively.

128

4.5 Summary

A stereological method has been used to analyze the distribution of WC/Co and

WC/WC interfaces in WC-Co composites. For WC/Co interfaces, it was found that

0110

facets m

prisma

boundaries,

planes m

90° about

prismatic and 0001 basal facets dominate the populations. These two special

ake up ~80 % of the WC/Co interfaces. The aspect ratio (base-to-height ratio of

tic plane) increases as the carbide volume fraction increases. For WC/WC

the prismatic and basal facets were also prominent. The basal and prismatic

ake up ~60 % of the entire grain boundary population. Grain boundaries with a

]0110[ make up 11~14 % of the population and boundaries with 30° about

[0001] make up 2~3 % of the population in straight angular grades (grades A~G),

respectively.

4.6 References

Observations of

. Kim and G.S. Rohrer, “Geometric and Crystallographic Characterization of

C Surfaces and Grain Boundaries in WC-Co Composites”, Interface Sci., 12, 19 (2004)

. Pang, H.M. Miller, P. Wynblatt, A.D. Rollett, and

[1] D.M. Saylor and G.S. Rohrer, “Determining Crystal Habits from

Planar Sections”, J. Am. Ceram. Soc., 85, 2799 (2002)

[2] C.-S

W

[3] D.M. Saylor, B.S. El-Dasher, Y

G.S. Rohrer, “Habits of Grains in Dense Polycrystalline Solids”, J. Am. Ceram. Soc., 87,

724 (2004)

129

[4] T. Sano, C.-S. Kim, and G.S. Rohrer, “Shape Evolution of SrTiO3 Crystals During

Coarsening in a Titania-Rich Liquid”, J. Am. Ceram. Soc., in press

[5] D.M. Saylor, B.L. Ad d G.S. Rohrer, “Measuring the Five Parameter Grain

Boundary Distribution From Observations of Planar Sections”, Metall. Mat. Trans., 35A,

1981 (2004)

[6] D.M. Saylor, B.S. El Dasher, A.D. Rollett, and G.S. Rohrer, “Distribution of Grain

Boundaries in Aluminum as a Function of Five Macroscopic Parameters”, Acta Mater.,

52, 3649 (

ams, an

2004)

87 , 670 (2004)

[8] M.G. Kendall and P.A.P. Moran, Geometrical Probability

[7] D.M. Saylor, B.S. El-Dasher, T. Sano, and G.S. Rohrer, “Distribution of Grain

boundaries in SrTiO3 as a Function of Five Macroscopic Parameters”, J. Am. Ceram. Soc.,

, Hafner Publishing

Powder Metall., 5, N

tific A

[12] A.V. Shatov, S.A. Firstov, nad I.V. Shatova, “The Shape of WC Crystals in

Cemented Carbides”, Mater. Sci. Eng. A, 242, 7 (1998)

Company, New York, 1963

[9] S.I. Wright and R.J. Larsen, “Extracting Twins from Orientation Imaging

Microscopy Scan Data”, J. Microscopy, 205, part 3, 245 (2002)

[10] D.N. French, “The Microstructure and Microproperties of WC-Co Alloys”, Int. J.

o. 3, 47 (1969)

[11] J. Gurland, “New Scien pproaches to Development of Tool Materials”, Int.

Mater. Rev., 33, No. 3, 151 (1988)

130

[13] S. Kim, S.-H. Han, J.-K. Park, and H.-E. Kim, “Variation of WC Grain Shape with

Carbon Content in the WC-Co Alloys during Liquid-phase Sintering”, Scripta Mater., 48,

635 (20

7] D. Wolf, “Correlation between Structure, Energy, and Ideal Cleavage Fracture for

ymmetrical Grain Boundaries in FCC Metals”, J. Mater. Res., 5, 1708 (1990)

8] J. Gurland and J.T. Norton, “Role of the Binder Phase in the Cemented Carbide-

Cobalt Alloys”, Trans. Metall. Soc. AIME, 194, 1051 (1952)

[19] A. Henjered, M. Hellsing, H.-O. Andren, and H. Norden, “Quantitative

Microanalysis of Carbide/carbide Interfaces in WC-Co-base Cemented Carbides”, Mater.

Sci. Technol., 2, 847 (1986)

[20] M.U. Farooq and U. Klement, “EBSD characterization of carbide-carbide

boundaries in WC-Co composites”, J. Microscopy, 213, 306 (2004)

03)

[14] A. Morawiec, “Misorientation-angle Distribution of Randomly Oriented

Symmetrical Objects”, J. Appl. Cryst., 28, 289 (1995)

[15] P. Wynblatt, M. Takashima, “Correlation of Grain Boundary Character with

Wetting Behavior”, Interface Science, 9, 265 (2001)


Phys. Stat. Sol. A, 61, 97 (1980)

[1

S

[1

131

Chapter 5

Finite Element Simulations of Fracture Strength

in WC-Co Composites

This chapter describes a finite element method (FEM) for predicting the

mechanical properties of WC-Co composites. A brittle fracture criterion has been

assumed to estimate the fracture strength of these materials. Both the digital grain maps

constructed on the basis of direct microstructural measurements (AFM maps, such as

those in Fig. 3.3(c) and OIM maps, such as those in Fig. 3.2 (b)) have been used as input.

In the first section, the simulation method using the object-oriented finite (OOF) element

code will be explained. The results of yield strength (YS) and transverse rupture strength

(TRS) calculations using real microstructures are presented in section two. By comparing

the results of calculations with measurements, we are able to select a fracture energy that

o repr The

calibrated simulation can then be used to calculate the properties of hypothetical

microstructures; this is discussed in section three.

5.1 Simulation methods

Object-oriented finite element analysis (OOF) was developed at National Institute

of Standards and Technology (NIST, Gaithersburg, MD). As described in Chapter 2, this

allows the simulation t oduce the observed transverse rupture strength.

132

program is designed to investigate the stress, stain and failure behavior in complex

microstructures. The program performs two-dimensional thermo-elastic computations

under mechanical and/or thermal loads. In this work, the responses to the combined loads

(mechanical + thermal) were calculated for the real WC-Co microstructures. For the

some of hypothetical microstructures, only mechanical loads were applied to estimate the

fracture strength. It has been found that while this simplified approach using only

mechanical loads leads to quantitatively different answers, the trend is unchanged and it

anges that are of interest in the hypothetical icrostructures.

ing m opy (OIM) and atomic force

procedures were used as input. Crystallographic orientations and elastic constants of each

hase (carbide and binder) were incorporated to model the anisotropic materials

properties. The crystallographic orientations of the WC crystals were obtained by

electron backscattered patterns (EBSPs), in the form of Euler angles (φ1, Φ, φ2). Table 5.1

[1, 2] presents the elastic constants of the WC and Co phases. Since the crystallographic

easured, the bulk properties (Young’s modulus,

Co, and Poisson’s ratio, νCo) have been used instead of the anisotropic stiffness (C11, C22,

C33, C12 13 23

out, how

is only these relative ch m

Digital grain maps from orientation imag icrosc

microscopy (AFM) that were used for the previously described microstructural analysis

p

orientations of the Co phase were not m

E

, C , and C ) of Co. In other words, the carbide phase was modeled as a

hexagonal material, while the binder phase was treated as an isotropic material. It turns

ever, that the isotropic assumption for Co does not affect the results significantly.

Tests using random assignments of Co orientations and actual stiffnesses yielded the

same results as tests using isotropic elastic constants. The consistency of the results from

133

the two elastic models presumably derives from the high concentrations of residual

stresses at sharp edges of the binder phase.

(a) (b)

program fail under a

Table 5.1 : Elastic constants of WC and Co [1, 2]. (a) Microstructural property (hexagonal

symmetry). (b) Bulk property (isotropic). E denotes Young’s modulus and ν denotes Poisson’s ratio.

As described in Chapter 2, the finite elements in the OOF

Griffith-like criterion [3],

AL ijijεσγ212 ≤∆ (5.1)

in which γ is the surface energy of the cracked interface, ∆L is the edge length, A is the

area, σij is the stress, and εij is the strain of a element, respectively. Therefore,

specifications of appropriate surface energies (or fracture energies, γ) are important to

simulate mechanical failures. Since there are four possible crack paths (transgranular

along WC/WC boundaries, transgranular along WC/Co interfaces, intergranular in WC

134

crystals, and intergranular in the Co) [4], it is necessary to define different fracture

energies for each case. Moreover, the transgranular fracture energies vary with the

crystallographic misorientations. Currently, it is not feasible to obtain the correct fracture

ork, therefore, we assume that cracks always initiate in the

transgranular fracture of

Fischmeister’s [5]. They reported that the fracture

simplify the description of OOF

, a digital image from AFM has been used as an input (see Fig. 5.1). Based

used for the OOF program. Using

performed an OOF (version 1.1.17)

at this point (crack initiation point) is considered to be

the yield strength (YS). Thus, the key to this analysis is to find the first crack initiation

energy distributions as a function of crystallographic orientation and/or misorientation in

WC-Co composites. In this w

WC phase (intergranular along WC/WC grain boundaries and transgranular in WC

grains), and that fracture energies (γ) for the intergranular and

the carbide phase are constant [5]. This assumption is supported by the theoretical

fracture energy models of Sigl and

energies of the WC phase are significantly lower than those of the WC/Co interface

and/or Co phase.

It has been found that thermal loads are important for determining the fracture

energy of the composites (see Section 5.2). However, to

simulation, an example (from grade A) is used that is subject only to a mechanical load.

While the actual calculations for WC-Co composites were performed using OIM maps, in

this example

on the microstructural data and the direct measurement technique described earlier (see

Section 3.2.3), a grain map is constructed and this was

a sub-program called ‘ppm2oof’ (version 1.1.27), the corresponding finite element mesh

can be generated as shown in Fig. 5.1(d). We then

simulation that allowed us to calculate the stress and strain at the point of fracture

according to Eq. (5.1). The stress

135

point as the mechanical load is incrementally increased. Fig. 5.1(e) and Fig. 5.1(f)

represent stress (stress invariant 1, σ11+σ22+σ33) distributions at the fracture point for

e-stress condition. Note that

x- and y- simulations.

[6] (see details in

Section 2.2.2). Even though it is possible that there is a very thin Co layer at the

carbide/carbide boundary, the layer was ignored in all calculations. This amounts to

saying that even if there is a thin layer of Co at WC/WC boundaries, it has a brittle and

hard character in comparison to bulk Co. In other words, it is assumed that the

‘skeletonized carbide’ hypothesis is acceptable for the mechanical property calculations.

elongation tests. The fractured elements are colored black (surrounded by circles). All of

the simulations in this thesis were carried out under the plan

the stress distributions for the x- and y- simulations are different, and that the fractured

elements are also different. All of the fracture strength calculation results presented in

this thesis are averages of the

One thing that should be mentioned is that all the simulation results discussed in

the later sections are based on the ‘skeletonized carbide’ hypothesis

136

ig. 5.1 : Example of OOF simulations under mechanical loads. (a) OIM IPF map. (b) AFM

topograx-elongation.

(a) (b)

(f)

(c) (d)

(e) F

phy map. (c) Grain map. (d) Finite element mesh. (e) stress invariant 1 distribution along (f) stress invariant 1 distribution along y-elongation. (see text for details).

137

Euler angles in OOF software

The Euler angles used in the OOF code are different from the conventional

definition of Bunge. OOF’s Euler angle (α, β, γ) definition is as follows [3]:

1) Rotate β about the z-axis

2) Rotate α about the new rotated y-axis

3) Rotate γ about the new rotated z-axis

After some matrix algebra, a general conversion from Bunge to OOF Euler angles

can be reached as:

Φα = (5.2a)

[ ]360),90(mod 2φβ −= (5.2b)

]360),270mod[( 1φγ −= (5.2c)

where mod is a function that returns the positive remainder, and φ1, Φ, and φ2 are Bunge

Euler angles defined in Chapter 2 (see Section 2.2.1 i)).

5.2 Results from real microstructures using brittle fracture model

The finite element analysis has been completed for specimen grades A, B, C, and

E. These particular grades are good examples for comparing the influence of the basic

microstructural parameters, grain size and contiguity (equivalently, volume fraction), on

strength. Effects of grain size can be investigated using grades A and B (different grain

138

size and same volume fraction), and effects of carbide volume fraction are estimable by

grades B, C, and D (similar grain size and different volume fraction). Table 5.2

mmarizes the microstructural parameters and measured transverse rupture strength

(TRS) for these four grades. All simulations in this thesis using real microstructures are

based on these four specimen grades.

su

Table. 5.2 : Summary of the microstructural and mechanical properties for grades A, B, C, and E.

was carried out based on the OIM maps of these four samples.

o produce reliable results, three independent sections of each specimen grade have been

analyzed. Each section (cropped from a larger map) contains approximately 600 to 900

C grains and is discretized using ~180,000 to 210,000 triangular elements. To transfer

the orientation information in an OIM map to OOF code, we have redefined the colors of

individual carbide grains according to their grain identification number (grain ID);

different red, green, and blue (R, G, B) values were assigned in each grain. (R, G, B)

values were specified in the order of (0, 0, 0) (pure black), (0, 0, 1), (0, 0, 2), … , (0, 0,

All the calculation

T

W

139

255) (pure blue), (0, 1, 0), (0, 1, 1), … , (0, 1, 255), (0, 2, 1), … as grain ID increases. Fig.

5.2 shows an example of input image (from specimen grade A) for OOF simulations. The

grain ID starts from the upper-left corner.

(a)

(b) (c) Fig. 5.2 : Conversion of colors for OOF simulations (grade A). (a) Original OIM IPF map, (b) new RGB map, and (c) grain map created in OOF code. See text for details.

140

First, to test the performance of the OOF program, we have calculated the

effective modulus (composite modulus) of each grade. In Fig. 5.3, calculated composite

moduli (based on 3 sections per grade) are compared to experimentally measured moduli.

The error bars indicate the standard deviations in the results from the three sections. It is

clear that the calculated moduli for all grades exhibit slightly lower values than the

measured data. These systematic negative deviations might be because the Co binder is

not pure and, therefore, doesn’t have the same modulus as pure Co, which was assumed

in the c

alculation. We can be sure that the binder in the composites contains dissolved W

and C and this might increase the modulus by a small amount. Thus, it is assumed that

the actual modulus of the Co in the composites is slightly higher than that of pure Co.

While the calculations do show systematic differences, it should be emphasized that the

absolute values of the errors fall in the range of 1.33 to 2.27 %.

Fig. 5.3 : Comparison of the calculated and measured modulus for each grade. Calculated moduli show systematic negative deviations.

141

∆T=-

1400° during y-

condition C range, the

al expansion coefficients

in the calculations were α11,WC=5.2×10-6/K and α33, WC=7.3×10-6/K for the carbide phase,

and αCo=1.4×10-5 /K for the binder phase [7]. Here, we assume that the thermal

expansion coefficients of the WC and Co phases are constant at all temperatures. An

example of an OOF simulation result is given in Fig. 5.4. The distributions of stress

invariant 1 ([Pa]) and elastic energy density (arbitrary unit, [a.u.]) are illustrated under a

combined load of ∆T=-1400 °C and 0.1 % y-elongation.

Since the surface energies (or fracture energy, or the two crack paths

(transgranular along WC/WC boundaries and intergranular in WC crystals) are not

available, we calculated the yield strength (YS) as a function of assumed fracture

energies. It was found that the calculated YS is linearly proportional to γ1/2. The linear

dependence of yield strength on γ1/2 can be derived from the Griffith-like brittle fracture

criterion (Eq. 5.1). Since the area of a element is proportional to edge length of the

element (

For the calculation of the fracture strength of composites, a thermal load of

C was applied with a plane stress and fixed (fix (y) the top and bottom

elongation and fix (x) the right and left during x-elongation simulations) boundary

s. Because the composites are sintered in the 1400 to 1600 °

quenching parameter was taken to be ∆T=-1400 °C. The therm

γ) f

LA ∆∝ ), fracture criterion equation can be written as:

LL

Aijijijij ∆εσ

∆εσγ

21

212 ∝≤ (5.3)

Lijij ∆εσγ41

∝ (5.4)

142

Here, the linear relationship between stress and strain in the elastic range (σ=Eε)

reduces Eq. (5.4) to . This is a plausible explanation for the variation of the

ield strength with fracture energies. Based on the relationship between yield

), the yield strengths at all values of the fracture

single calculation of yield strength at a specific

ess invariant 1 ([Pa]) and elastic energy density ([a.u.]) distributions (from grade A).

energy density under a combined load of ∆T=-1400 °C and 0.1 % y-elongation.

Using a brittle fracture assumption, the calculated yield strength was taken as the

fracture strength. As described before, we ignore the plastic process. It should be noted

that we do not expect the fracture energies that describes fracture in WC-Co composites

to be similar to a WC surface energy. We are assuming a brittle process, even though it is

known that plastic processes occur during the failure at these materials. Thus, our

assumption is that all of the complete energy dissipation process that occurs during

σγ ∝2/1

calculated y

( σγ ∝2/1strength and fracture energy

energy can be easily determined from a

fracture energy.

4.0e9 7.7e8

Fig. 5.4 : Str

(a) Stress invariant 1 under a combined load of ∆T=-1400 °C and 0.1 % y-elongation, (b) elastic

1.5e9

Pa

(b)(a) 1.1e7

143

fracture can be represented by a single average value, γ. Since typical surface energy is of

order 1 J/m2, we expect this g to be considerably higher.

Since transverse rupture strength (TRS) is usually used as a measure of strength in

WC-Co composites, the effective fracture energy in the carbide phase was calibrated so

that the calculated YS of four specimen grades agreed with the measured TRS values.

hroughout this thesis, fracture strength is treated as the same quantity as TRS. With the

approach described above, we compute the best-fit value of γ, using results for all three

sections for each grade. The effective fracture energy was therefore empirically

determined to be ~ 49 J/m2. This result is remarkably consistent with the reported fracture

energy value, ~ 50 J/m2 [5]. The calculated results are compared with measured TRS data

in Fig. 5.5. The results are averaged based on the analyses of three sections for each

grade and the error bars denote standard deviations arising from the variation from

section to section. The calculated strengths are consistent with the measured data. All the

ergy assumption provides a useful way to predict the fracture strength of WC-Co

compos

T

errors fall within the range of 6 %. This indicates that the constant effective fracture

en

ites regardless of the grain size and volume fraction. It suggests that, using the

effective fracture energy (γ=49 J/m2), the strength is plausibly approximated irrespective

of the volume fraction. These results indicate that grain size has a negative effect and

carbide volume fraction has a positive effect on strength, as is the case with hardness.

Furthermore, it appears that coarse-grains are more detrimental to strength than to

hardness.

144

Fig. 5.5 : Comparison of the calculated and measured fracture strength. Error bar denote standard deviation (in sections) of calculated fracture strength. All the calculated fracture strengths are based on the analysis of three sections per each grade.

To examine the effect of the resolution of the digital images on the results of the

simulations, we calculated the strength of composites using AFM images as input. There

are two disadvantages of using AFM images for OOF simulations. First, an automatic

assignment of physical properties (orientations, stiffness, and thermal expansion

coefficients) is not possible. In previous simulations, all of the physical parameters were

given by automatic assignments. Second, large area simulations are unavailable; whereas

IM section contains 600~900 carbide grains, one AFM section includes only

AFM maps have an

ecall that the resolutions are 0.04 (grades B, C, and E) to 0.12 (grade A) µm/pixel in the

AFM images, but they are 0.15 (grades B, C, and E) to 1.0 (grade A) µm/pixel in the OIM

one O

120~150 carbide grains. On the other hand, simulations from

advantage in that the resolutions of the AFM images are better than those in OIM maps.

R

145

scans.

to for predicting realistic

acture energies in the carbide phase.

For a more realistic calculation of the fracture strength, it is necessary to develop

a plastic fracture model that includes the binder plasticity. Currently, however, it is not

feasible to develop such a model, since available finite element methods are applicable

only to the elastic region. The model described above simply averages all of the energy

dissipation processes into effective fracture energy.

5.3 Results from hypothetical microstructures using brittle fracture model

the model calibrated to calculate the fracture strength of eal materials, the

icrostructures were then calculated to investigate the

In other words, the AFM images have the advantage of accurate boundary

positions, but have the disadvantage of limiting the maximum size of the computational

section. We compared the calculated strengths from AFM and OIM images and the

results turned out to be very similar (not shown). Those results demonstrate that OIM

maps can be used to produce accurate simulations even though the boundary positions are

not resolved as well as they are in the AFM images. The comparison was made using

results from simulations that considered only mechanical loads. It is noteworthy that the

simulations performed with only mechanical loads can produce a result that is also

consistent with the measured strength data. However, the calibrated effective fracture

energy under mechanical loads alone was found to be significantly small than the

expected value, ~50 J/m2. This means that there must be some pre-existing stresses

within materials due to cooling and thermal loads are important

fr

With r

fracture strengths of hypothetical m

146

effects of microstructural features (the contiguity of the WC phase, the shape of WC

grains, and the WC grain size distribution) on fracture strength. To properly evaluate the

individual influences of these microstructural features, simple hypothetical

microstructures have been generated either from simulations (Monte-Carlo seeding

method) or from real microstructures of other composite materials. Because the

contiguity, angularity, size ratio,

res with square-shaped

rystals are illustrated. These microstructures were designed to have different contiguities

ide volume fraction, grain

microstructures of the hypothetical materials could be controlled to produce nearly

independent variations in the microstructural parameters, the effects on the fracture

strength were clearly revealed. Although some of the hypothetical microstructures

presented below are very simple, they clearly demonstrate that the fracture strength can

be potentially enhanced through the appropriate control of the microstructural parameters.

Because it has been found that the calculated strength under a mechanical load alone also

exhibits a consistent result with measured data, only mechanical loads have been applied

to the simulations using hypothetical microstructures for

and size distributions. For the simulations using textured hypothetical microstructures, a

combined load was applied.

5.3.1 Contiguity

i) Hypothetical microstructures from simulations

In Fig. 5.6 (a)~(c), three simple hypothetical microstructu

c

while maintaining the carb size, size distribution, and

147

angularity. Here, color contrast was randomly assigned to distinguish individual grains.

These microstructures were created by the following algorithm:

1) Determine the desired microstructural features; contiguity (C), average grain size ( d ),

standard deviation (σ), maximum inimum grain size, and carbide volume fraction

(f

and m

ld contiguity

) Generate a random integer within the range of maximum and minimum grain sizes.

4) Calculate the probability for finding the size of the grain in step 1) ac g to

Gaussian distribution:

c).

2) Calculate the o

3

cordin

⎥⎦

⎤⎢⎣

⎡ −−= 2

2

2)(exp

21)(

σπσddrp (5.5)

enerate a random number (0 to 1)

) Generate random integers (x- and y-positions within domain) to locate the grain

created

5) G

6) If the calculated probability (p(r)) in step 4) is greater than the random number (0 to 1)

in step 5), go to step 7). Otherwise, go to step 3).

7

in step 3).

8) Calculate the new contiguity using all grains in the domain

9) Adjust the contiguity of the microstructure. When the old contiguity is less than the

predetermined contiguity in step 1), if the new contiguity is greater than the old one,

locate the grain, else discard the grain and go to step 3). When the old contiguity is

148

greater than the predetermined contiguity, if the new contiguity is less than the old one,

locate the grain, otherwise, discard the grain and go to step 3).

10) Calculate the carbide volume fractions. If the calculated carbide volume fraction is

greater than the pre-determined fraction (fc), stop. Otherwise, go to step 3).

With this algorithm, it is possible to control the contiguity with preserving nearly

same quantities of other microstructural features. The values of fc and d are the most

important parameters to be kept constant, since they have the most significant effect on

the properties. In Fig. 5.6, the carbide volume fraction was set to be 0.9 and average grain

edge length was ~1.5 (arbitrary unit, [a.u.]) in each microstructure. Because the

contiguity of real WC-Co composites is ~0.59 with carbide volume fraction of 0.9

(specimen grades A and B), the values selected for the simulation were 0.59, 0.35 and

0.02. The same simulation conditions were used as for the real WC-Co microstructures:

plane stress and fixed boundary conditions. But, for the crystallographic properties of

WC phase, averaged values of the bulk Young’s modulus (=714 GPa) and Poisson’s ratio

(=0.19) were used. Since we already tested the effect of crystallographic anisotropies

under a mechanical load (see Section 5.2.1 i)), which turned out to be negligible,

assigning average moduli and Poisson’s ratios will not strongly influence the results.

Fig. 5.6 (d)~(i) shows the stress invariant 1 (hereafter SI1, in units of [Pa]) and

elastic energy density (hereafter EED, in units of [a.u.]) distributions with different

contiguities (mechanical load alone, 0.1 % y-elongation). It is apparent that stresses and

ored energies are more concentrated in the WC phase in the high contiguity

microstructure. The dissipation of stresses through WC phase is prohibited by the Co

st

149

phase in the low contiguity microstructure. This trend is also found in the data in Fig. 5.7.

The average stresses on each phase were averaged from five independent sections created

using the algorithm described above. While the average stress on the composites are

nearly the same, the average stress ratio of the carbide to binder phase (average stress of

WC/average stress of Co) increases as the contiguity increases. The contiguity effect is

more obvious when we compare the calculated fracture strengths under mechanical loads

in Fig. 5.8. Error bars stand for the standard deviations in five different sections. The

estimated fracture strength increases by ~7 % as the contiguity decreases from 0.59 to

thing that needs to be noticed is that the width of grain size distributions in

he low contiguity microstructure has a

ider distribution whereas the high contiguity microstructure shows a narrower

distribu

0.02. One

these microstructures decreases with the contiguity because we used a probabilistic

approach to make hypothetical microstructures. T

w

tion. The normalized standard deviations (σ/ d , standard deviation/average grain

size) are ~0.5, ~0.7, and ~0.8 with the contiguities of 0.59, 0.35, and 0.02. Since it has

been found that the wide distribution has a negative effect on fracture strength (see

Section 5.3.4), the effects of contiguity in the strength would presumably have been

larger if the hypothetical microstructures had the same grain size distribution.

150

y-elongation), fixed and plane stress boundary conditions. (a) Low contiguity (0.02) structure, (c) high contiguity (0.59)

e, (d)~(f) stress invariant 1 distributions (same scale) for (a), (b), and (c) es, respectively, (g)~(i) elastic energy density distributions (same scale) for (a), (b),

re concentrated in the carbide phase.

(a) (b) (c)

(d) (e) (f)

1.6e9

1.0e8

Pa

2.1e6

4.0e4(g) (h) (i)

Fig. 5.6 : Examples of hypothetical microstructures with their stress invariant 1 ([Pa]) and elastic energy density ([a.u.]) distributions. Simulations performed under mechanical load alone (0.1 %

microstructure, (b) medium contiguity (0.35) micromicrostructurmicrostructurand (c) microstructures, respectively. As the contiguity increases, more SI1 and EED a

151

Fig. 5.7 : Average stresses in the composites, carbide and binder phases with different contiguities (0.1% y-elongation, mechanical load alone, plane stress and fixed boundary

one section (same section shown in Fig. 5.6).

conditions). Average stress in the carbide phase increases as contiguity increases. Results from

Fig. 5.8 : Plot of the calculated fracture strength ([a.u.]) with different contiguities. Mechanical load with plane stress and fixed boundary conditions was applied. Results are averages of five sections. Error bars represent standard deviations in sections. Hypothetical microstructure with low contiguity has a higher fracture strength.

152

ii) Hypothetical microstructures from real microstructures

Since the hypothetical microstructures used in previous section are rather simple

compared to the real WC-Co composites, modified microstructures were created based on

icrostructures with same grain size, and carbide volume fraction, but

WC grain boundary.

4) Calculate the carbide volume fraction.

) If the calculated carbide fraction is greater than the pre-determined carbide fraction in

step 1), then go to step 2). Otherwise, return the converted Co phase in step 3) to the

original WC/WC boundary and stop.

The shortcoming of this method is that it is not feasible to create a higher

contiguity microstructure than the original one; some WC/WC grain boundaries are

always eliminated. Hence, the pre-determined carbide volume fraction must be smaller

than the fraction in real microstructure. Furthermore, it is not possible to fully control the

contiguity. Once the newly calculated carbide volume fraction is less than the pre-

determined fraction, the process is stopped. Therefore, in every trial to create a

real microstructures in order to test the reproducibility of the contiguity effect on strength.

Hypothetical m

different contiguities were made by the following procedure:

1) Specify the desired carbide volume fraction.

2) Randomly select a WC/

3) Convert the selected boundary into Co.

5

153

hypothetical microstructure, the final contiguity cannot be fixed, and it varies within a

certain range.

OIM sections of specimen grade B ( cf =0.8992, d =1.65 µm) were chosen as the

original microstructure because they have the highest contiguity. From these sections, we

reated microstructures with carbide volum fractions of ~0.69 that were analogous to

specim

c e

en grade E ( f =0.6949, c d =1.40 µm). In other words, the hypothetical structure

created from grade B has same microstructural features as grade E

E are different, but it can be easily modified by changing the size of computational

domains in the OOF simulations. Fig. 5.9 contains the hypothetical and real grade E

microstructures with their responses (EED distribution) to a mechanical load (0.1 % y-

elongation). They have a same carbide fraction (=0.69) but different contiguities, 0.32

(real) and 0.06 (hypothetical). Here, the EED distribution for only the carbide phase is

illustrated for the purpose of visualization (Fig. 5.9 (c) and (d)). It is also seen that the

smaller than in a high contiguity (real) sample. The calculated fracture strengths are 2.53

and 2.71 GPa for the high and low contiguity microstructures, respectively. Note that

experimentally measured TRS is 2.57 GPa and standard deviation is ~0.09 GPa for

, but different

contiguity. Of course, the average carbide grain sizes of the hypothetical and real grades

energy propagation through the WC in the low contiguity (hypothetical) microstructure is

specime

to the measured value, the strength for the hypothetical grade (Fig. 5.9 (a)) is beyond

n grade E. While the calculated strength for the real grade E (Fig. 5.9 (b)) is close

E

the range of the standard deviation.

154

Simulations performed under mechanical load alone (0.1 % y-elongation), fixed and plane stress

(carbide phase, same scale) for low and high contiguity microstructures (a) and (b), respectively.

method. It

rned out none of the high contiguity samples have better performance in strength than

the low contiguity ones. Fig. 5.10 shows another example. Here, a hypothetical low

contiguity grade C ( =0.83, C=0.28) was created and compared with a real sample

=0.83, C=0.50). In this case also, a high elastic energy density distribution in the

carbide phase was observed in the real microstructure and the estimated strengths were

2.80 and 2.93 GPa for the real and hypothetical specimens, respectively (the measured

5.0e5

1.0e5

(a) (b)

(c) (d)

Fig. 5.9 : Examples of hypothetical microstructures with their EED ([a.u.]) distributions.

boundary conditions. (a) Low contiguity (0.06) microstructure (hypothetical grade E, from real grade A), (b) high contiguity (0.32) microstructure (real grade E), (c) and (d) EED distributions

Low contiguity sample shows low EED.

We tested several examples using the ‘WC/WC boundary removal’

tu

cf

( cf

155

TRS for grade C is 2.84 GPa). The reason for the improved performance of the low

contiguity samples is illustrated schematically in Fig. 5.11. When the contiguity is high,

the load is transmitted through the carbide phase, which is stiffer. When the contiguity is

low, the more compliant binder carries more of the load.

Fig. 5.10 : Examples of hypothetical microstructures with their elastic energy density (a.u.) distributions. Simulations performed under mechanical load alone (0.1 % y-elongation), fixed and plane-stress boundary conditions. (a) Low contiguity (0.28) microstructure (hypothetical grade C, from real grade A), (b) high contiguity (0.50) microstructure (real grade C), (c) and (d) stress invariant 1 distributions (carbide phase, same scale, a.u.) for low and high contiguity microstructures (a) and (b), respectively. Low contiguity sample shows low energy densities.

(a) (b)

(c) (d)

5.0e5

1.0e5

156

chematic illustration of stress distributions of high (left) and low (right) contiguity

.3.2 Angularity


The effect of angularity on strength was explored using hypothetical images with

completely dispersed WC particles (C=0). Fully dispersed microstructures have been

selected because it is difficult to control angularity while maintaining constant finite

contiguity. It is, however, already verified that low contiguity microstructures exhibit

better strengths. Thus, it can be assumed that we are testing the effect of angularity in the

best possible (C=0) of constant contiguity.

rostructures were created with relatively low carbide volume fraction (=0.55)

ls (see Fig. 5.12 (a)~(c)). The locations of WC

high stress

low stress

WC

WC

WC

WC

Fig. 5.11 : Smicrostructures. While high stresses are concentrated in WC phase in a high contiguity sample, the propagation of stresses are blocked by Co phase in a low contiguity sample.

5

Mic

to avoid the impingement of carbide crystals. To simplify the situation, we divided

microstructures into binary categories; square shapes represent high angularity particles

and circles represent low angularity crysta

157

crystals were randomly selected. Orientations in the samples with the square shapes were

either randomly assigned (square 1, see Fig. 5.12 (a)) or fixed (square 2, see Fig. 5.12 (b)).

In Fig. 5.12 (a)~(i), the SI1 (carbide and binder phases) and EED distributions (carbide

phase only) are shown with their corresponding microstructures. Simulations were

carried out under mechanical load alone (0.1 % y-elongation), fixed and plane stress

boundary conditions. It was found that the samples with square shapes experience higher

stresses and elastic energies than those with circular shapes. This is because the stresses

and elastic energies are more easily redistributed in the microstructures containing curved

interfaces. Also, it has been observed that the microstructure made up of squares with

fixed orientations (square 2) shows less SI1 and EED in the carbide phase than the

microstructures with random orientations (square 1). This fact presumably arises from

differences in binder free path distributions. As can be seen in Fig. 5.12 (a) and (b), the

binder free path distribution in square 1 is wider than that in square 2. This suggests that a

uniform distribution of binder free paths leads to an increase in the strength of the

composites.

The calculated fracture strengths are plotted in the Fig. 5.13. Only a mechanical

load has been applied and the results are averages of five independent sections (error bars

indicate standard deviations in results from different sections). Data shows that fracture

strength increases by ~3 % as the shape changes from square 1 to circle, and the stress

ratio (average stress in WC /average stress in Co) decreases by ~6% as the shape changes

from square 1 to circle (not shown). Based on the results from simple hypothetical

microstructures discussed above (square 1, square 2, and circle), it can be concluded that

the low angularity sample has an advantage over the high angularity structure.

158

Fig. 5.12 : Examples of hypothetical microstructures with their SI1 ([Pa]) and EED ([a.u.]) distributions. Simulations performed under mechanical load alone (0.1 % y-elongation), fixed and plane stress boundary conditions. (a) Square microstructure with random orientations (square 1),

distributions (same scale) for square 1, square 2, and circle, respectively, (g)~(i) EED distributions (same scale) for square 1, square 2, and circle, respectively.

(b) square microstructure with a fixed orientation (square 2), (c) circle microstructure, (d)~(f) SI1

159

160

ot of the calculated fracture strength ([a.u.]) with different angularities (or shapes). averages of five sections. Error bar

Fig. 5.13 : PlResults are esent standard deviations in sections.

e shape shows a higher fracture strength.

ed, reducing angularity increases strength. To test

e generality of this assertion, we made high angularity hypothetical microstructures

distributions ([a.u.]) under a 0.1 % y-elongation mechanical load. The W-Ni-Fe

composites have approximately ellipsoidal particles (W) in the Ni-Fe matrix, which is a

good model for low angularity microstructures; the ellipsoidal W crystals can be treated

s reprHypothetical microstructure with a circl

ii) Hypothetical microstructures from real microstructures of other composites

The results described above suggest that in a fully dispersed composite with a

relatively low carbide volume fraction

th

using the ‘WC/WC boundary removal’ method (see Section 5.3.1 i)) and compared their

mechanical response to a low angularity microstructure with a similar contiguity derived

from a different composites (W-Ni-Fe alloy, courtesy of A.D. Rollett, Carnegie Mellon

University). Fig. 5.14 shows these hypothetical microstructures with their EED

as carbide particles and Ni-Fe matrix can be treated as the Co binder. In the W-Ni-Fe

alloy, particle volume fraction and contiguity are calculated to be ~0.79 and ~0.16,

respectively, so hypothetical angular microstructures of same carbide fraction (0.79) and

contiguity (0.16) are generated from a real WC/Co microstructure using the ‘boundary

moval’ method described in Section 5.3.1 ii).

e

hypothetical m No

microstructures. F

nd (b), respectively. We tested several

other sections (low and high contiguity structure with fc=0.79 and C=0.16), and the

calculated fracture strength in high angularity microstructures from the ‘boundary

removal’ method is greater by 0.1 GPa on average. It was found that the dihedral angles

at junctions where Co meets a WC bicrystal are important in determining the crack

initiation and streng (see Fig. 5.15).

re

Fig. 5.14 (a) is the original OIM micrographs of grade B, and (c) is th

icrostructure after WC/WC boundaries were selectively removed.

visible difference was found in EED distributions for these low and high angular

racture strengths were determined to be 2.81 and 2.89 GPa for the low

and high angularity sample shown in Fig. 5.14 (a) a

th

161

162

Fig. 5.14 : Examples of hypothetical microstructures with their EED ([a.u.]) distributions. Simulations performed under mechanical load alone (0.1 % y-elongation), fixed and plane stress

microstructure (W-Ni-Fe alloy, courtesy of A.D. Rollett, Carnegie Mellon University), (c) input for the OOF simulation of high angularity micremoved, from (a)), (d) input for the OOF sim

boundary conditions. (a) High angularity microstructure (original, grade A), (b) low angularity

rostructure (WC/WC boundaries were selectively ulation of low angularity microstructure (same as

(b)), (e)~(f) EED distributions (same scale) for high and low angularity microstructures, respectively. Color contrasts in (a)~(b) are randomly assigned to distinguish individual grains.

The fact that the high angularity structure is better than the low angularity

structure seems to be diametrically opposed to the preceding results, but recall that the

situations are different. The simulations in the previous section (Section 5.3.2 i)) were

carried out under zero contiguity (fully embedded), whereas the current calculations are

conducted under finite contiguity (C=0.16). Also, the carbide volume fractions are

different: ~0.55 for the previous and ~0.79 for current sections. Therefore, it allows us to

conclude that the angularity has two (positive and negative) effects on the strength

depending on the carbide fraction and contiguity. When the carbide fraction is small

enough and the contiguity is very small, low angularity increases the strength. However,

in commercial WC-Co composites, where the carbide faction is 0.7~0.9, the spherical or

ellipsoidal particles are considered to be detrimental to the strength.

elongation simulation.

Fig. 5.15 : Schematic illustration of the stress concentration at sharp edge of WC phase in a y-

high stress concentrated

WC

WC

Co binder

163

164

5.3.3 Aspect ratio


The influence of aspect ratio on the strength is tested by simple hypothetical

microstructures using the ‘random location’ method described in section 5.3.2 i). Here,

the hypothetical microstructures were again made up of fully dispersed structures.

Carbide shapes are modeled as squares and rectangles for fixed aspect ratios of 1:1, 1:2,

and 1:4. Fig. 5.16 illustrates an example of hypothetical microstructures, SI1 (carbide and

binder phases), and EED (carbide phase only) distributions under 0.1 % y-direction

mechanical strain. Average grain sizes, angularity (each particle consists of four edges),

contiguity (=0), and carbide fraction (=0.55) are the same for all of the samples in Fig.

5.16 (a)~(c). High stress and energy concentrations were found along the elongated

grains parallel to the direction of the mechanical load (see Fig. 5.16 (f) and (i)). The

calculated fracture strengths are compared with different aspect ratios in Fig. 5.17, which

shows that the fracture strength decreases by ~4 % as the aspect ratio increases from 1:1

density exhibit wider

istribution. The stress and elastic energy is directly transmitted through elongated

carbide particles that are parallel to the loading direction. Although the model structures

discussed here are very simplified, they clearly show that equiaxed (aspect ratio of 1:1)

to 1:4. In this plot, the strengths have been averaged from five different sections and error

bars represent the standard deviations in sections. The reason for the better performance

of the microstructures with equiaxed carbide crystals is schematically illustrated in Fig.

5.18. When the aspect ratio is large, the stress and elastic energy

d

shapes can provide better reinforcements than elongated (aspect ratios of 1:2 and 1:4)

shapes.

distributions. Simulations performed under mechanical load alone (0.1 % y-elongation), fixed and

SI1 distributions (same scale) for (a)~(c) structures, respectively, (g)~(i) EED distributions (same

0.0

0.0

Pa

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

1.0e9

8.0e5

Fig. 5.16 : Examples of hypothetical microstructures with their SI1 ([Pa]) and EED ([a.u.])

plane stress boundary conditions. (a) Square microstructure with aspect ratio of 1:1, (b) rectangle microstructure with aspect ratio of 1:2, (c) rectangle microstructure with aspect ratio of 1:4, (d)~(f)

scale) for (a)~(c) structures, respectively.

165

Fig. 5.17 : Plot of the calculated fracture strength ([a.u.]) with different aspect ratios. Results are

microstructure with equiaxed shape has a higher fracture strength.

The effect of aspect ratio on the strength of composites is schematically illustrated

in Fig. 5.18. High stresses and elastic energies are easily transmitted through the

elongated carbide grains that are paralle

averages of five sections. Error bars represent standard deviations in sections. Hypothetical

l to the loading direction. Less stresses and elastic

nergies were found in the carbide grains elongated perpendicular to the loading direction.

In other words, in samples with high aspect ratios, the stress and elastic energy density

distributions is wider than those in samples with equiaxed carbide shapes, and this is

detrimental to the fracture strength of WC-Co composites.

e

166

Fig. 5.18 : Schematic illustration of the stress concentrations with different aspect ratios in a y-elongation simulation. High stresses propagate through vertical WC particles (red).

5.3.4 Size distribution


The effect of size distributions on strength was investigated in a way that is

similar to the studies of angularity and aspect ratio. We assume that the carbide shape is

square, the orientation is random, and the carbide particles are fully dispersed (C=0).

Again, a relatively low carbide volume fraction (=0.55) was assigned in order to avoid

Based on the calculations for fracture strengths (averages of five independent sections) of

three different size distributions, it has been found that the fracture strength decreases by

high stress

low stress

impingement of the WC particles. An example of hypothetical microstructures with

different square edge size ratio (1:1, 1:2, and 1:4) is given in Fig. 5.19 (a)~(c). The

number of small and large crystals in Fig. 5.19 (b) and (c) is identical. EED (carbide

phase only, [a.u.]) distributions under 0.1 % y-direction strains are illustrated in the

bottom (Fig. 5.19 (d)~(f)). From these images, we conclude that high stresses and elastic

ergies pass through the large grains rather than small grains (see Fig. 5.19 (d)~(f)). en

167

~3 % a

hypothetical microstructures with their EED ([a.u.]) distributions.

(a) Square microstructure with edge size ratio of 1:1, (b) square icrostructure with edge size ratio of 1:2, (c) square microstructure with edge size ratio of 1:4,

(d)~(f) EED distributions (same scale) for (a)~(c) structures, respectively.

s the size ratio increases from 1:1 to 1:4 (see Fig. 5.20). Therefore, it is concluded

that a homogeneous size distribution will certainly benefit for the strength of WC-Co

composites.

The reason for improved strength in samples with a uniform size distribution is

illustrated schematically in Fig. 5.21. When the size distribution is wider, the load can be

easily transmitted through the larger carbide grains, while the stresses in the smaller

grains are low. On the other hand, when the size distributions is narrower (or uniform),

the stresses and elastic energies are evenly distributed, thus, the improved strength of

composites is expected.

8.0e5

0.0

(a) (b) (c)

(d) (e) (f)

Fig. 5.19 : Examples of Simulations performed under mechanical load alone (0.1 % y-elongation), fixed and plane stress boundary conditions. m

168

Fig. 5.20 : Plot of the calculated fracture strength ([a.u.]) with different size ratios. Results are

microstructure with homogeneous distribution (1:1 size ratio) has a higher fracture strength.

elongation simulation. High stresses propagate through large WC particles (red).

averages of five sections. Error bars represent standard deviations in sections. Hypothetical

Fig. 5.21 : Schematic illustration of the stress concentrations with different size ratios in a y-

high stress

low stress

169 169

Note that all of the simulations for the aspect ratio and size distribution (section

5.3.3 i), and 5.3.4 i)) were executed with a zero contiguity and a low carbide fraction

0.55), which might be thought to be a highly impractical situation. It is, however,

considered that these constraints in the simulations do not alter the results. If we have had

a finite contiguity and high carbide fraction, the influences of aspect ratio and size

distribution would have been much more apparent, because stresses and stored energies

in the carbide phase are able to be easily spread out through the adjacent carbide crystals.

5.3.5 Orientation and Misorientation

The effects of orientation and misorientation textures on fracture strength were

studied using hypothetical microstructures. We first examined the effects of orientation

and misorientation on some simple microstructures, and then tested the orientation

texture effects using hypothetical WC-Co microstructures.

i) Simple examples

Fig. 5.22 illustrates three simple microstructures with (001)C//[001]S (the (001)

plane of crystal is parallel to the [001] direction of sample) orientation. Here, this will be

referred to [001] orientation texture. A small square crystal is embedded in another large

, α2=0.99×10 , and α3=1.01×10 /K).

(=

matrix grain. The three microstructures differ in the misorientation between the dispersed

and matrix crystals. Here, the crystals are designated as pseudo-cubic materials whose

lattice parameters are cubic, but whose thermal expansion coefficients are different

(orthorhombic characters, α1=1.00×10-5 -5 -5

170

(a) (b) (c) Fig. 5.22 : Schematic illustration of (001)C//[001]S orientation texture samples (pseudo-cubic materials) with different misorientations. One crystal is embedded in another matrix crystal. (a)

0° misorientation, (b) 45° misorientation, and (c) 90° misorientation. Red and pink arrows denote a specific direction, [100], for the embedded and matrix crystal, respectively. Section normals

1] direction of sample reference frame.

ructures (misorientations of

° to 90°) were prepared and simulated with a thermal load of ∆T=-1000° with free and

s the misorientation between

two crystals increases, the stress and elastic energy associate with the thermal load

increases (minimum at a misorientation of 10°, and maximum at a misorientation of 90°).

point the [00

Employing the same idea, a series of simple microst

10

plane stress boundary conditions. In the simulation, the assigned physical parameters

were: Young’s modulus=500 GPa, Poisson’s ratio=0.33, thermal expansion coefficient

α1=1.00×10-5, α2=0.99×10-5, α3=1.01×10-5/K, and specimen width=20 µm. Fig. 5.23 and

Fig. 5.24 show SI1 ([Pa]) and EED ([a.u.]) distributions of these microstructures.

Although the difference in thermal expansion coefficients is approximately 1~2% along

the x-, y-, and z-directions, the stress and energy resulting from those thermal

anisotropies clearly show the influence of misorientations. A

171

applied. All plots are same scale. As misorientation angle increases, the stress caused by different

10° 20°

50° 60°

30° 5.5e7

-5.2e7

Pa

40°

70° 80° 90°

Fig. 5.23 : SI1 ([Pa]) distribution of [001] orientation texture samples (pseudo-cubic materials)

with different misorientations. ∆T=-1000°C with free and plane stress boundary conditions was

thermal expansion coefficients increases.

172

Fig. 5.24 : EED ([a.u.]) distribution of [001] orientation texture samples (pseudo-cubic materials)

with different misorientations. ∆T=-1000°C with free and plane stress boundary conditions wasapplied. All plots are same scale. As misorientation angle increases, the stored energy caused by different thermal expansion coefficients increases.

To generalize the misorientation effect on residual stresses and energies, we have

10° 20° 30°

40° 50° 60°

70° 80° 90°

1.3e3

0

tested another set of microstructures. In this case, nine square crystals reside in the

computational domain and they all have different orientations (see Fig. 5.25 (a)~(d)).

Among them, the first two (Fig. 5.25 (a) and (b)) has (001)C//[001]S have orientation

173

texture, and the last two (Fig. 5.25 (c) and (d)) do not have any orientation texture

(random orientations). Note that the orientation distribution in each pair of

microstructures is preserved; the only thing that is changed is the arrangement of the

crystals. In other words, rearranging the crystals in sample (a) makes structure (b), and

rearranging the crystals in (c) makes microstructure (d). The microstructures have been

constructed such that the average misorientation is maintained as low as possible in the

sample (a) and (c), and as high as possible in the specimen (b) and (d), under fixed

orientation distributions. The numerical values in the textured sample (Fig. 5.25 (a) and

(b)) indicate orientations with respect to a certain reference frame, so the misorientation

o crystals is the difference of the corresponding orientations (all tilt

The stress and energy distributions under a thermal load (∆T=-1000°C) are

depicted in Fig. 5.25 (e)~(l) with the corresponding microstructures. The bondary

conditions and physical properties used in these simulations were exactly the same as

those described in the preceding section. All SI1 and EED distributions are plotted in a

same scale. Low SI1 and EED distributions were observed in samples with a small

average misorientation angle (compare the first and second columns, third and fourth

columns in Fig. 5. 25). As expected, the residual stress and energy are more concentrated

(the boundary

etween 50 and 60 in Fig. 5.25 (f) and (j)), the intensities of stress and elastic energy

were almost negligible compared to those at a high angle boundary. The same trend was

between tw

boundaries). The average misorientations were calculated to be 20°, 35°, 28°, and 50° for

the microstructures of (a), (b), (c), and (d) in Fig. 5.25.

along the high angle boundaries (the boundaries between 0 and 50, 10 and 60, and 10 and

80 in Fig. 5.25 (f) and (j)). On the other hand, at the low angle boundary

b

174

found

ough the average misorientation of (b) is smaller than that of (c), the residual stress and

in (b). For example, the maximum elastic energy

tation textures. It is generally

ccepted that the orientation texture reduces the stored energy in polycrystalline materials,

nge of possibilities depending on the associated misorientations (see

in the randomly oriented microstructures (the third and fourth columns in Fig.

5.25). The results demonstrate that it is feasible to diminish the stored energies and

increase the fracture strength by rearranging crystals with the same orientation

distributions.

The effect of the orientation distribution was also investigated (compare the

second and third columns in Fig. 5.25). Recall that microstructure (b) is has an average

misorientation of 35° and microstructure (c) has an average misorientation of 28°. Even

th

energy in (c) are much larger than those

densities are 1.2×103, and 2.6×103 ([a.u.]), and the average energy densities are 3.49, and

6.39 ([a.u.]) for (b) and (c), respectively. This means that the three dimensional mismatch

in the lattice will produce more stress and strain energy, and it suggests that these stresses

and energies are able to be reduced by controlling the orien

a

but there is a wide ra

Fig. 5.26). We have also calculated the fracture strength in these microstructures under

thermal load of ∆T=-1000°C. Although the results are based on simple microstructures,

the data in Fig. 5.27 show that the fracture strength can be greatly increased by increasing

the orientation texture and reducing misorientations.

175

Fig. 5.2 ffects of orientation and

C S

(e)~(h) SI1 distribution for microstructures (a)~(d), (i)~(l) EED distribution for microstructures

See text for details.

0 10 20

30 40 50

60 70 80

0 40 80

50 60 10

20 70 30

(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j) (k) (l)

4.3e7

-5.6e7

Pa

2.6e3

0

5 : E misorientation distributions on stress ([Pa]) and elastic energy density ([a.u.]) distributions. (a) Microstructure with a (001)C//[001]S orientation texture and a small misorientation angle, (b) microstructure with a (001) //[001] orientation texture and a high misorientation angle, (c) microstructure with random orientations and a small misorientation angle, (d) microstructure with random orientations and a high misorientation angle,

(a)~(d), respectively. Samples with the (001)C//[001]S orientation texture display low stress and energy. Also, samples with a small misorientation angle show low stress and energy distributions.

176

Fig. 5.26 : St ergy density variation tation and misorien extures. Generallymaterials with a high orientation texture display a low energy density distrib on than materiawith random orientations. The range of stored energy density distribution is much wider in samples with orientation textures.

ored en with orien tation tuti

, ls

Fig. 5.27 : Estimated fracture strength in materials (Fig. 5. 25 (a)~(d)). The sample with (001)C//[001]S orientation texture and a small misorientation shows the highest performance.

177

ii) Hypothetical microstructures

In the previous section, we verified that strength could be enhanced by orientation

and misorientation texture. To employ this idea to WC-Co composites, model

microstructures with orientation textures are proposed. Here, we neglect the geometrical

constraints to orientations. In real WC-Co materials, the orientation is related to the

geometry (shape) in the two-dimensional sections. For example, the <0001> axis of a red

crystal (usually observed in a triangular shape) in an OIM IPF map (see Fig. 4.6) must be

nearly parallel to the [001] sample axis. In

correlated with the two-dimensional shapes that appear in planar sections. In the

simulations described here, this correlation will be removed.

To impose some textured orientations, we first generate a set of random Euler

angles and calculate the angle between a certain crystal axis and a fixed sample axis. Fig.

5.28 illustra le (∆) between [001] sample axis and <0001> crystal

axis.

Fig. 5.28 : deviation angle ( ) between the [001] sample axis and a <0001> crystal axis.

other words, the orientations are strongly

tes the deviation ang

∆

sample

crystal

[001] sample [001] crystal

deviation angle (∆)

178

)/exp(

We then compute the probability of finding a crystal with the randomly assigned

Euler angles according to the following formula:

β∆−=p (5.6)

that determines the intensity (or degree) of orientation texture, and p is the probability of

finding a crystal with a deviation angle of ∆. We used β=2, 10, and 25 to construct

qualitatively high, medium, and low orientation textured microstructures, respectively.

Fig. 5.29 shows the probability (p) as a function of the orientation texture intensity

constant, β, and difference in angles, ∆, between the crystal and sam

where ∆ (in degrees) is the angle between a crystal axis and a sample axis, β is a constant

ple axes. It is clearly

seen that in the high texture (β=2), the probability of finding a crystal with a deviation

angle (∆) greater than 10° is nearly 0. We then generate another random number (0 to 1)

and if the random number is less than the computed probability (Eq. (5.6)), the pre-

assigned Euler angles are accepted, otherwise, they are discarded. The process is repeated

for all of the carbide crystals. In this manner, hypothetical microstructures with

(001)C//[001]S ([001] texture), (100)C//[100]S ([100] texture), and (111)C//[111]S ([111]

texture) orientation textures were created. Here, for convenience, the three-digit notation

(instead of the four-digit notation) is used for the plane normal and axis directions of

hexagonal materials.

179

Fig. 5.29 : Probability of finding a crystal as a function of texture intensity factor (β) and

Composite moduli were first calculated using these microstructures with

hypothetical microstructures derived from a section of grade A. There are nine distinct

models, three (high, medium, and low) [001] textures, three [100] textures, and three [111]

deviation angle (∆) from a sample axis.

hypothetical orientations (see Fig. 5.30). The results presented here are all based on

textures. In Fig. 5.30, moduli of the [001] and [100] textured samples were calculated by

averaging the results from the x- and y-elongation simulations. However, the modulus of

[100] textured sample was measured separately because the responses along the x- and y-

directions are fairly different. This difference results from the anisotropy of elastic

constants of WC. Remember that the stiffness along c-axis (C33) in the WC crystal is

much higher than other directions (C11=720, C33=972, C44=328, C12=254, C13=267 GPa).

Therefore, in the [001] textured sample, the stiffness along the x- ([100] of sample) and

y- ([010] of sample) directions is the averaged by the elastic constants of the axes

180

perpendicular to c-axis (see Fig. 5.31 (a)). Likewise, the anisotropy in the stiffness

doesn’t affect the modulus of the x- and y-directions in the [111] textured sample. On the

other hand, in the [100] textured sample, the prismatic facets are mostly perpendicular to

x-direction and basal planes are facing the y-direction of the sample reference frame (see

Fig. 5.31 (b)). This explains the difference in modulus along the x- and y-directions in the

[100] textured sample. Fig. 5.30 describes the calculated modulus of the nine different

materials. The dotted horizontal line denotes the experimentally measured modulus of

real (no orientation texture) grade A. It is clear that as the degree of the texture decreases

(β=2 25), the calculated modulus approaches to the measured value. The modulus of

the [001] textured sample is always lower than the measured value since the stiffest axis

(c-axis) was excluded in the calculation. In the case of the [100] textured sample, the

modulus along c-axis direction (y-direction of sample) is invariably higher than the

measured quantity, and the modulus along the direction perpendicular to c-axis (x-

direction of sample) is lower than the experimental value. The modulus of [111] texture

is nearly e as the measured quantity, which means the response is almost isotropic sam

along the x- and y-direction of sample reference frame.

181

Fig. 5.30 : Calculated modulus with different orientation textures (from grade A). Dotted line indicates experimentally measured modulus. As the degree of texture decreases, the calculated modulus approaches to the measured value. See text for details.

(a) (b) Fig. 5.31 : Schematic illustration of the orientation in (a) [001] texture, and (b) [100] texture. The

ely represent the orientations, not the actual shape of hypothetical microstructures.

prismaticbasal prismatic basal

shapes mer

182

The SI1 and EED were then computed for these hypothetical microstructures (see

Fig. 5.32 and 5.33). A thermal load of ∆T=-1400°C was applied with free and plane stress

boundary conditions. The same scales were used for SI1 and EED to visualize the relative

distributions. The distributions in the real microstructure (no texture) are illustrated at the

bottom of Figs. 5.32 and 5.33. It is apparent that the sample with high [001] texture

shows reduced stresses and energy densities. As the intensity of texture decreases, the

stress and energy distributions resemble the original distribution.

183

[001] high [1 [111] high

ig. 5.32 : SI1 ([Pa]) distributions with different orientation textures (from grade A). A thermal

load of ∆T=-1400°C was applied with free and plane stress boundary conditions. Same scale was used. The high [001] texture (upper-left) shows a low stress distribution.

[001] medium

[001] low

[100] medium

[100] low

[111] medium

[111] low

00] high

original (no texture)

3.8e

-4.5e9

Pa

F

184

[001] high

[001] low

[100] high

[100] low

[111] high

[111] low

[001] medium [100] medium [111] medium

used. The high [001] texture (upper-left) shows a low energy density distribution.

original (no texture)

6.3e6

0

Fig. 5.33 : EED ([a.u.]) distributions with different orientation textures (from grade A). A thermal

load of ∆T=-1400°C was applied with free and plane stress boundary conditions. Same scale was

185

The fracture strength was calculated under a combined thermal and mechanical

load in six sections of the nine (three with [001] texture and three with [100] texture,

from grade A). Simulations were carried out using the same conditions as in the real

microstructure (fixed and plane stress boundary conditions, and ∆T=-1400°C, see Section

5.2.3). Here, we tested the best ([001] texture) and worst (y-direction in the [100] texture)

cases. In other words, we took the softest (the best case, x- and y- in [001] texture) and

stiffest (the worst case, y- in [100] texture) axes for the strength calculations. In Fig. 5.34

(a), the strength of the [001] textured sample is the average of the x- and y-directions and

the strength of the [100] textured sample was obtained by y-direction simulations. The

results show that the [001] textured samples clearly have a higher strength; the dotted line

indicates the strength of the real section (1.83 GPa, from grade A). Even in the

calculations of the stiffest axis (y-direction of [100] texture), the approximated strength is

not worse than that of real microstructure. The result indicates that the strength can be

potentially increased by approximately 2 (the worst case) to 13 (the best case) % by

texturing the sample (see Fig. 5.34 (b)).

186

(a)

(b) Fig. 5.34 : The calculated (a) fracture strength and (b) fracture strength increase (%) with

eal icrostructure (grade A).

different orientation textures. Simulations were performed along the softest ([001] texture) and stiffest (y-direction in [100] texture) axes. The dotted line in (a) represents the strength of rm

187

Up to now, the effects of orientation texture on mechanical properties were

presented by a single section from grade A. To generalize these observations, we expand

the calculation to other grades (grades B, C, and E). Three different sections have been

analyzed for each grade. The calculated fracture strength increase (%) is given in Fig.

5.35. The results are based on 72 (four different grades × three sections for each grade ×

six (three [001] and three [100] y) different textures) independent simulations. The range

of the calculated fracture increase was calculated to be approximately -2 to 13 %. At all

mes, the maximum strength was found in the high [001] textured samples, and the

minimum strength was found in the y-direction of high [100] textured samples. In the y-

direction of high [100] textured sample, sometimes the calculated strength was lower

than that of real sample; note that the minimum change in specimen grade C is -2 % (not

shown in Fig. 5.35). Thus, the orientation texture does not always promote the strength

along a certain direction, but it is considered that, generally, orientation textures with low

misorientation angles enhance the strength of WC-Co composites. For more detail

investigations, three-dimensional simulations need to be carried out, but, currently, this is

not feasible in the OOF code.

Finally, it was observed that the relative maximum increment of the strength for

different specimen grades is linear with the contiguity of the samples (see Fig. 5.36). It

means the effect of WC/WC boundary misorientations is greater in high contiguity

samples.

ti

188

Fig. fracture in the hyp res of grades A, B, C, and E. Three sections were averaged for each grade. Maximum increment was found to be ~13% ([001] texture in the hypothetical A and B), and the minimum was approximated to be -2% (not shown, y-direction of [100] texture in the hypothetical C).

5.35 : The calculated strength increase (%) othetical microstructu

Fig. 5.36: The calculated maximum fracture strength increase as a function of contiguities. Three sections were averaged for each grade. Error bar indicates the standard deviation in sections. As the contiguity increases, the relative fracture strength increases.

189

5.4 Summary

A brittle fracture model for th ngth of WC-C as been

developed using finite element analysis (FEM). The model assumes that a crack initiates

in the brittle carbide phase and that the fracture strength is proportional to the yield

strength. When we use ~49 J/m2 as a constant surface energy under a combined load

(thermal and mechanical), the brittle fracture model can approximate the fracture strength

(transverse rupture strength). The calculate hows

good agreement with the experimental data. With this brittle fracture model calibrated to

calculate the fracture strength of real materials, the strength of hypothetical

microstructures was then calculated. The most important factors appear to be the

connectivity (contiguity) of the carbide skeleton, the orientation texture, and the

misorientation texture. The angularity, the crystal size distribution, and the aspect ratio of

the WC also influence strength, but these pa ters are subordinate

texture.

5.5 References

[1] M. Lee and R.S. Gilm e Crystal Elastic Constants of Tungsten

e fracture stre o composites h

d fracture strength using this model s

rame to contiguity and

ore, “Singl

Monocarbide”, J. Mater. Sci., 17, 2657 (1982)

[2] A. Kelly and G.W. Groves, Crystallography and Crystal Defects, Longman Inc.,

London, 1973

190

[3] W.C. Carter, S.A. Langer and E.R. Fuller Jr., The OOF Manual (version 1.0.8.6),

National Institute of Standards and Technology (NIST), (2001); also available online at


[4] L.S. Sigl and H.E. Exner, “Experimental Study of Mechanics of Fracture in WC-Co

Alloys”, Metall. Trans. A, 18A, 1299 (1987)

[5] L.S. Sigl and H.F. Fischmeister, “On the Fracture Toughness of Cemented Carbides”,

Acta Metall., 36, No. 4, 887 (1988)

[6] S. Bartolucci and H.H. Schlosin, “Plastic Deformation Preceding Fracture in

Tungsten Carbide-Cobalt Alloys”, Acta Metall. Mater., 14, No. 3, 337 (1966)

[7] W.M. Stoll and A.T. Santhanam, Kirk-Othmer Encyclopedia of Chemical

Technology, 4 ed., 4, John Wiley and Sons, Inc., 861, 1992 th

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Chapter 6

Summary

This thesis focused on microstructure-property relationships in WC-Co

composites and had three main components: quantitative microstructural measurements,

measurements of the interface character distribution, and the development of a model to

predict strength. Nine different specimen grades (seven straight angular grades, one

on-angular grade, and one alloyed plate-like grade) with the average carbide

volume fractions of 0.7 to 0.9 were used

throughout this work.

For the microstructural characterization of these composites, the microscopic

features were measured directly to avoid the inaccuracies and limitations of the

conventional linear-intercept method. Geometric and crystallographic information was

obtained using atomic microscopy (AFM) and orientation imaging microscopy (OIM).

The composites studied in this work had very homogeneous microstructures. In all grades,

the log normal distributions of carbide grain size and binder free path resemble the

Gaussian distribution with normalized standard deviations of 0.55 to 0.60. It was found

is a function of only the volume fraction of the carbide phase and it is

ge of 0.7 to 0.9. The average number of

etermined by the carbide grain size and contiguity (equivalently,

straight n

grain sizes of one to seven microns and carbide

that the contiguity

strongly linear to the volume fraction in the ran

vertices per carbide grain was observed to be roughly constant; it varies from six to seven

in the angular grades. The measured hardnesses and fracture toughnesses of these

composites are mainly d

192

carbide volume fraction). Empirical expressions involving these two major parameters,

the carbide grain size and contiguity, show a good agreement with the experimental

values of the hardness and toughness.

A stereological method has been used to quantify the distributions of WC/Co

interfaces and WC/WC grain boundaries. The surfaces of WC in contact with Co are

mostly composed of 0110 prismatic and 0001 basal facets; 80 % of the WC/Co

interfaces belong to one of those two habits within 5° tolerance angle. Approximately

two thirds of the populations of these boundaries are prismatic planes and one third are

basal planes, which leads the aspect ratio (base-to-height) of the prism shape to be 1.18 to

1.52, depending on the specimen grades. In the WC/WC grain boundaries, it was also

found that the prismatic and basal planes dominate the populations. The population of

these special facets is 60 % of the total boundary within a tolerance angle of 5°. Among

them, there are two special boundaries, the 90° ]0110[ and 30° [0001] boundaries, which

make up 11 to 14 % and 2 to 3 % of the total boundaries in the straight angular grades.

The populations of these special boundaries in for the straight non-angular grade and the

plate-like grade are approximately 7 % (90°), 1 % (30°), and 6 % (90°) and 7 % (30°),

respectively.

The fracture strengths of WC-Co composites have been calculated using a two-

dimensional finite element method (FEM) based on a Griffith-like brittle fracture criteria.

The model approximates the fracture strength as the yield strength. The calculated

fracture strength with the calibrated single fracture energy (~49 J/m2) in the carbide phase

shows a consistency with the measured transverse rupture strength. In calibrating the

model, we found that the residual stress and strain resulting from a thermal load (cooling

193

down from the sintering temperature) is important to determine the fracture energy of the

carbide phase. With this brittle fracture model, hypothetical microstructures were

examined to investigate the effects of microstructural features on the strength of WC-Co

composites. In Fig. 6.1, the effects of microstructural features are schematically

illustrated. Here, the magnitudes of arrows indicate the extent of effect on the strength.

The average grain size and volume fraction of the carbide phase mainly determine the

fracture strength; this is also the case for hardness and fracture toughness. The contiguity

and the texture of the carbide phase are the next most important parameters. It is

t the strength can be increased by as much as ~7 % by reducing the

ous size

the strength by ~3 to 4 %.

In this work, we didn’t analyze the influence of the fraction of 90° and 30° special

boundaries on mechanical properties. It has been reported that low energy boundaries can

be used to improve mechanical properties, such as intergranular stress cracking [1-5],

creep behavior, cavitation characteristics [6-9], fracture toughness [10, 11], cold work

embrittlement [12], and weldability [13]. In WC-Co composites, it is anticipated that the

mechanical properties would also be improved by increasing the fraction of special

boundaries. Remarkably, the special boundary fraction was nearly constant in the

commercial grades analyzed in this work. To include the effect of special boundaries,

icated simulations need to be conducted; finer boundary meshes must be

predicted tha

contiguity from 0.59 to 0.02 and/or by adopting a high degree of orientation and

misorientation texture. The carbide shape and size distribution effects on the strength

were also studied. It was observed that the equiaxed grain shape and homogen

distribution can potentially increase

more compl

created and different fracture energy values must be specified. Also, for realistic

194

calculations to include the effects of grain boundary distributions, a three-dimensional

simulation should be performed. This still remains a challenge for the future work.

ig. 6.1 : Schematic illustration of the effects of microstructural features on the fracture strength ide shapes are highly simplified. Different color contrasts for the texture

effect represent different orientations.

grain

carbide fraction

texture

contiguity angularitysize

distribution

aspect

small

large

high

low

high low

low wide elongated

gth

size ratio

low high

high narrow equiaxed fracture stren

Fof composites. Carb

195

6.1 References

[1] G. Palumbo and K.T. Aust, “Localized Corrosion at Grain Boundary Intersections in

-Dependence of Intergranular Corrosion in

Aust et al., “Grain Boundary Design and Control for

Character

[5] D.C. Crawford and G.S. Was, “The Role of Grain Boundary Misorientation in

Intergranular Cracking of Ni-16Cr-9Fe in 360 °C Argon and High Purity Water”, Metall.

Trans. A, 23A, No.4, 1195 (1992)

[6] J. Don and S. Majumdar, “Creep Cavitation and Grain Boundary Struture in Type

304 Stainless Steel”, Acta Metall., 34, No.5, 961 (1987)

[7] D.P. Field and B.L. Adams, “Interface Cavitation Damage in Polycrystalline Copper”,

Acta Metall., 40, No.6, 1145 (1992)

[8] V. Thaveeprungsriporn and G.S. Was, “The Role of Coincidence Site Lattice

Boundaries in Creep of Ni-16Cr-9Fe at 360 °C”, Metall. Trans. A, 28A, No.10, 2101

(1997)

High Purity Nickel”, Scripta Metall., 22, No.6, 847 (1988)

[2] G. Palumbo and K.T. Aust, “Structure

High Purity Nickel”, Acta Metall., 38, No.11, 2343 (1990)

[3] G. Palumbo, P.J. King, and K.T.

Intergranular Stress-Corrosion Resistance”, Scripta Metall., 25, No.8, 1775 (1991)

[4] P. Lin, G. Palumbo, U. Erb et al., “Influence of Grain Boundary

Distribution on Sensitization and Intergranular Corrosion of Alloy 600”, Scripta Metall.,

33, No.9, 1387 (1995)

196

[9] E.M. Lehockey, G. Palumbo, P. Lin et al., “On the Relationship between Grain

Boundary Character Distribution and Intergranular Corrosion”, Scripta Mater., 36, No.10,

1211 (1997)

d T. Watanabe, “Fracture Toughness and Brittle-Ductile Transition

Contro

. Palumbo, and P. Lin, “Improving the Weldability and Service

Perform

[10] L.C. Lim an

lled by Grain Boundary Character Distribution (GBCD) in Polycrystals”, Acta

Metall., 38, No.12, 2507 (1990)

[11] E.M. Lehockey, G. Palumbo, K.T. Aust et al., “On the Role of Intercrystalline

Defects in Polycrystal Plasticity”, Scripta Mater., 39, No.3, 341 (1998)

[12] E.M. Lehockey, G. Palumbo, and P. Lin, “Grain Boundary Structure Effects on

Cold Work Embrittlement of Microalloyed Steels”, Scripta Mater., 39, No.3, 353 (1998)

[13] E.M. Lehockey, G

ance of Nickel- and Iron-Based Superalloys by Grain Boundary Engineering”,

Metall. Trans. A, 29A, No.12, 3069 (1998)

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microstructural-mechanical property relationships in wc-co

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